| L(s) = 1 | + 10·3-s + 55·9-s + 2·13-s − 2·23-s + 21·25-s + 220·27-s − 2·29-s + 20·39-s + 34·43-s − 5·49-s − 2·53-s + 4·61-s − 20·69-s + 210·75-s + 38·79-s + 715·81-s − 20·87-s + 52·101-s − 44·103-s − 20·107-s − 18·113-s + 110·117-s + 14·121-s + 127-s + 340·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 5.77·3-s + 55/3·9-s + 0.554·13-s − 0.417·23-s + 21/5·25-s + 42.3·27-s − 0.371·29-s + 3.20·39-s + 5.18·43-s − 5/7·49-s − 0.274·53-s + 0.512·61-s − 2.40·69-s + 24.2·75-s + 4.27·79-s + 79.4·81-s − 2.14·87-s + 5.17·101-s − 4.33·103-s − 1.93·107-s − 1.69·113-s + 10.1·117-s + 1.27·121-s + 0.0887·127-s + 29.9·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{10} \cdot 7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(793.3298435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(793.3298435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{10} \) |
| 7 | \( ( 1 + T^{2} )^{5} \) |
| 13 | \( 1 - 2 T - 11 T^{2} - 64 T^{3} + 134 T^{4} + 1188 T^{5} + 134 p T^{6} - 64 p^{2} T^{7} - 11 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| good | 5 | \( 1 - 21 T^{2} + 249 T^{4} - 2228 T^{6} + 15438 T^{8} - 85166 T^{10} + 15438 p^{2} T^{12} - 2228 p^{4} T^{14} + 249 p^{6} T^{16} - 21 p^{8} T^{18} + p^{10} T^{20} \) |
| 11 | \( 1 - 14 T^{2} + 15 p T^{4} - 4248 T^{6} + 51218 T^{8} - 433780 T^{10} + 51218 p^{2} T^{12} - 4248 p^{4} T^{14} + 15 p^{7} T^{16} - 14 p^{8} T^{18} + p^{10} T^{20} \) |
| 17 | \( ( 1 + 33 T^{2} + 64 T^{3} + 622 T^{4} + 2048 T^{5} + 622 p T^{6} + 64 p^{2} T^{7} + 33 p^{3} T^{8} + p^{5} T^{10} )^{2} \) |
| 19 | \( 1 - 93 T^{2} + 239 p T^{4} - 157420 T^{6} + 4212650 T^{8} - 89413006 T^{10} + 4212650 p^{2} T^{12} - 157420 p^{4} T^{14} + 239 p^{7} T^{16} - 93 p^{8} T^{18} + p^{10} T^{20} \) |
| 23 | \( ( 1 + T + 63 T^{2} + 108 T^{3} + 2422 T^{4} + 2838 T^{5} + 2422 p T^{6} + 108 p^{2} T^{7} + 63 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 29 | \( ( 1 + T + 55 T^{2} + 8 T^{3} + 1724 T^{4} + 926 T^{5} + 1724 p T^{6} + 8 p^{2} T^{7} + 55 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 31 | \( 1 - 69 T^{2} + 3637 T^{4} - 153772 T^{6} + 5424330 T^{8} - 183076510 T^{10} + 5424330 p^{2} T^{12} - 153772 p^{4} T^{14} + 3637 p^{6} T^{16} - 69 p^{8} T^{18} + p^{10} T^{20} \) |
| 37 | \( 1 - 46 T^{2} + 4229 T^{4} - 148648 T^{6} + 9444306 T^{8} - 277558420 T^{10} + 9444306 p^{2} T^{12} - 148648 p^{4} T^{14} + 4229 p^{6} T^{16} - 46 p^{8} T^{18} + p^{10} T^{20} \) |
| 41 | \( 1 - 142 T^{2} + 13261 T^{4} - 920504 T^{6} + 50629058 T^{8} - 2308510452 T^{10} + 50629058 p^{2} T^{12} - 920504 p^{4} T^{14} + 13261 p^{6} T^{16} - 142 p^{8} T^{18} + p^{10} T^{20} \) |
| 43 | \( ( 1 - 17 T + 259 T^{2} - 2604 T^{3} + 23398 T^{4} - 162102 T^{5} + 23398 p T^{6} - 2604 p^{2} T^{7} + 259 p^{3} T^{8} - 17 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 47 | \( 1 - 201 T^{2} + 20689 T^{4} - 1415908 T^{6} + 74876702 T^{8} - 3573947014 T^{10} + 74876702 p^{2} T^{12} - 1415908 p^{4} T^{14} + 20689 p^{6} T^{16} - 201 p^{8} T^{18} + p^{10} T^{20} \) |
| 53 | \( ( 1 + T + 171 T^{2} - 320 T^{3} + 12256 T^{4} - 39794 T^{5} + 12256 p T^{6} - 320 p^{2} T^{7} + 171 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 59 | \( 1 - 542 T^{2} + 134613 T^{4} - 342120 p T^{6} + 2024504258 T^{8} - 141950970388 T^{10} + 2024504258 p^{2} T^{12} - 342120 p^{5} T^{14} + 134613 p^{6} T^{16} - 542 p^{8} T^{18} + p^{10} T^{20} \) |
| 61 | \( ( 1 - 2 T + 229 T^{2} - 448 T^{3} + 24710 T^{4} - 37596 T^{5} + 24710 p T^{6} - 448 p^{2} T^{7} + 229 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 67 | \( 1 - 214 T^{2} + 29669 T^{4} - 3166184 T^{6} + 280104482 T^{8} - 20394399300 T^{10} + 280104482 p^{2} T^{12} - 3166184 p^{4} T^{14} + 29669 p^{6} T^{16} - 214 p^{8} T^{18} + p^{10} T^{20} \) |
| 71 | \( 1 - 422 T^{2} + 86973 T^{4} - 11741176 T^{6} + 1174891314 T^{8} - 92867141316 T^{10} + 1174891314 p^{2} T^{12} - 11741176 p^{4} T^{14} + 86973 p^{6} T^{16} - 422 p^{8} T^{18} + p^{10} T^{20} \) |
| 73 | \( 1 - 189 T^{2} + 385 p T^{4} - 3135172 T^{6} + 295355718 T^{8} - 23357241406 T^{10} + 295355718 p^{2} T^{12} - 3135172 p^{4} T^{14} + 385 p^{7} T^{16} - 189 p^{8} T^{18} + p^{10} T^{20} \) |
| 79 | \( ( 1 - 19 T + 367 T^{2} - 4396 T^{3} + 54318 T^{4} - 480578 T^{5} + 54318 p T^{6} - 4396 p^{2} T^{7} + 367 p^{3} T^{8} - 19 p^{4} T^{9} + p^{5} T^{10} )^{2} \) |
| 83 | \( 1 - 549 T^{2} + 151517 T^{4} - 27252492 T^{6} + 3505741530 T^{8} - 335441550686 T^{10} + 3505741530 p^{2} T^{12} - 27252492 p^{4} T^{14} + 151517 p^{6} T^{16} - 549 p^{8} T^{18} + p^{10} T^{20} \) |
| 89 | \( 1 - 353 T^{2} + 73901 T^{4} - 11322316 T^{6} + 1378552978 T^{8} - 135568081158 T^{10} + 1378552978 p^{2} T^{12} - 11322316 p^{4} T^{14} + 73901 p^{6} T^{16} - 353 p^{8} T^{18} + p^{10} T^{20} \) |
| 97 | \( 1 - 469 T^{2} + 122857 T^{4} - 22205956 T^{6} + 3058897110 T^{8} - 331693035854 T^{10} + 3058897110 p^{2} T^{12} - 22205956 p^{4} T^{14} + 122857 p^{6} T^{16} - 469 p^{8} T^{18} + p^{10} T^{20} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.88482031003029753735817449909, −2.84538246886871332451251011633, −2.61333564658528531388995483925, −2.51196056958608098890369845593, −2.44864670179853595810420766013, −2.35070405047035503221761777005, −2.34990465974202247373986107622, −2.34372345705825321879420004277, −2.30971378703081680564548977606, −2.12064714627358985013608228902, −2.04167464774647387637826924099, −2.02278364392698026888763503013, −1.73927077859868322350411666010, −1.72407757454616935995952449787, −1.39047483801555103182149490615, −1.34995075441075808394433891654, −1.28834750348520380613960188724, −1.24825021437426138751784126651, −1.23310275781118950151963878321, −0.977788208870383573784361447104, −0.855130711577169740974292419302, −0.804030790202150393425481040144, −0.54066370389451240197192166366, −0.48278791996653411404751578706, −0.25813601760668573679468562529,
0.25813601760668573679468562529, 0.48278791996653411404751578706, 0.54066370389451240197192166366, 0.804030790202150393425481040144, 0.855130711577169740974292419302, 0.977788208870383573784361447104, 1.23310275781118950151963878321, 1.24825021437426138751784126651, 1.28834750348520380613960188724, 1.34995075441075808394433891654, 1.39047483801555103182149490615, 1.72407757454616935995952449787, 1.73927077859868322350411666010, 2.02278364392698026888763503013, 2.04167464774647387637826924099, 2.12064714627358985013608228902, 2.30971378703081680564548977606, 2.34372345705825321879420004277, 2.34990465974202247373986107622, 2.35070405047035503221761777005, 2.44864670179853595810420766013, 2.51196056958608098890369845593, 2.61333564658528531388995483925, 2.84538246886871332451251011633, 2.88482031003029753735817449909
Plot not available for L-functions of degree greater than 10.
