| L(s) = 1 | − 5-s + 3·7-s + 2·11-s − 5·13-s + 5·17-s − 7·19-s − 5·23-s − 25-s − 3·29-s + 7·31-s − 3·35-s − 6·37-s + 12·41-s − 8·43-s − 3·47-s + 49-s − 10·53-s − 2·55-s + 14·59-s + 61-s + 5·65-s − 4·67-s + 8·71-s − 7·73-s + 6·77-s − 7·79-s + 25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s − 1.38·13-s + 1.21·17-s − 1.60·19-s − 1.04·23-s − 1/5·25-s − 0.557·29-s + 1.25·31-s − 0.507·35-s − 0.986·37-s + 1.87·41-s − 1.21·43-s − 0.437·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s + 1.82·59-s + 0.128·61-s + 0.620·65-s − 0.488·67-s + 0.949·71-s − 0.819·73-s + 0.683·77-s − 0.787·79-s + 2.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438297158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438297158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 88 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 114 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 162 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 314 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 177 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.151351780896868803537957614361, −8.084434864127599446233812559273, −7.63561085502536896843565875864, −7.62565201392848534208393950920, −6.77939592309090475012557918763, −6.76822183873627090848629986429, −6.31442473907196185258306332949, −5.80128600826346724316156836406, −5.44008991167181008298849033925, −5.08555538358555783927809843297, −4.64677179246479093866380383172, −4.40839239828497161791199787234, −3.81688924256423231085219439691, −3.80743430366499973464708362708, −2.95900043919272169232512714004, −2.64027965529731933882660872674, −1.99153042970297041116046491294, −1.76974501336820173402615237861, −1.12830577068619285268915328276, −0.33248652120329347485811586158,
0.33248652120329347485811586158, 1.12830577068619285268915328276, 1.76974501336820173402615237861, 1.99153042970297041116046491294, 2.64027965529731933882660872674, 2.95900043919272169232512714004, 3.80743430366499973464708362708, 3.81688924256423231085219439691, 4.40839239828497161791199787234, 4.64677179246479093866380383172, 5.08555538358555783927809843297, 5.44008991167181008298849033925, 5.80128600826346724316156836406, 6.31442473907196185258306332949, 6.76822183873627090848629986429, 6.77939592309090475012557918763, 7.62565201392848534208393950920, 7.63561085502536896843565875864, 8.084434864127599446233812559273, 8.151351780896868803537957614361
