| L(s) = 1 | − 56·4-s + 490·7-s + 1.02e3·16-s − 6.00e3·19-s + 3.60e3·25-s − 2.74e4·28-s − 7.85e3·31-s − 406·37-s + 1.87e4·43-s + 1.46e5·49-s − 6.03e4·61-s + 3.58e3·64-s − 6.03e4·67-s + 1.84e5·73-s + 3.36e5·76-s + 7.87e4·79-s − 2.01e5·100-s + 5.85e5·103-s + 4.13e5·109-s + 5.01e5·112-s + 1.24e4·121-s + 4.39e5·124-s + 127-s + 131-s − 2.94e6·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 7/4·4-s + 3.77·7-s + 16-s − 3.81·19-s + 1.15·25-s − 6.61·28-s − 1.46·31-s − 0.0487·37-s + 1.54·43-s + 61/7·49-s − 2.07·61-s + 7/64·64-s − 1.64·67-s + 4.05·73-s + 6.67·76-s + 1.42·79-s − 2.01·100-s + 5.43·103-s + 3.33·109-s + 3.77·112-s + 0.0773·121-s + 2.56·124-s + 5.50e−6·127-s + 5.09e−6·131-s − 14.4·133-s + 4.55e−6·137-s + 4.38e−6·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.925911363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925911363\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 5 p^{2} T + p^{5} T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + 7 p^{3} T^{2} + 33 p^{6} T^{4} + 7 p^{13} T^{6} + p^{20} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 3604 T^{2} + 3223191 T^{4} - 3604 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 12460 T^{2} - 25782173001 T^{4} - 12460 p^{10} T^{6} + p^{20} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 584089 T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2697418 T^{2} + 5260069966275 T^{4} - 2697418 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 3003 T + 5482102 T^{2} + 3003 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 8382514 T^{2} + 28840029746547 T^{4} - 8382514 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 19845266 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 3927 T + 33769594 T^{2} + 3927 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 203 T - 69302748 T^{2} + 203 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 231705052 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4697 T + p^{5} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 116727080 T^{2} - 38973921030503649 T^{4} + 116727080 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 781349954 T^{2} + 435620280250289067 T^{4} + 781349954 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 1405123198 T^{2} + 1463254448257105803 T^{4} - 1405123198 p^{10} T^{6} + p^{20} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 30198 T + 1148569369 T^{2} + 30198 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 30197 T - 438266298 T^{2} + 30197 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 3592767500 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 92379 T + 4917698140 T^{2} - 92379 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 39385 T - 1525878174 T^{2} - 39385 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 3837977680 T^{2} + p^{10} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 62763994 T^{2} - 31177780611023351565 T^{4} - 62763994 p^{10} T^{6} + p^{20} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 17170432214 T^{2} + p^{10} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.28155351142116561439315337333, −9.738571622518777764430809801075, −9.105713196988070020110662839779, −9.035359296455537233954367865253, −8.721931433063530567172389068174, −8.547205162984101544439511941446, −8.483265875717866409645976109435, −7.86004872286152593505921259162, −7.64686319162157859929268942938, −7.52305291462404807913746577866, −6.91784798929355510636316461200, −6.31832827073590899144843151306, −6.03435809251748062215670728760, −5.57692483124641357850947097585, −4.94272085306200569905742017015, −4.78417563677086150185547734078, −4.75049121435581100521974186356, −4.20215300055554008749326490045, −4.13227855018354116295642424793, −3.45411691602373278205225265708, −2.28634691255403143932480141922, −2.02602167781240745596273655563, −1.79522513458509439042459382811, −0.871947835499704179834162005396, −0.47162285963257687313774658486,
0.47162285963257687313774658486, 0.871947835499704179834162005396, 1.79522513458509439042459382811, 2.02602167781240745596273655563, 2.28634691255403143932480141922, 3.45411691602373278205225265708, 4.13227855018354116295642424793, 4.20215300055554008749326490045, 4.75049121435581100521974186356, 4.78417563677086150185547734078, 4.94272085306200569905742017015, 5.57692483124641357850947097585, 6.03435809251748062215670728760, 6.31832827073590899144843151306, 6.91784798929355510636316461200, 7.52305291462404807913746577866, 7.64686319162157859929268942938, 7.86004872286152593505921259162, 8.483265875717866409645976109435, 8.547205162984101544439511941446, 8.721931433063530567172389068174, 9.035359296455537233954367865253, 9.105713196988070020110662839779, 9.738571622518777764430809801075, 10.28155351142116561439315337333
