| L(s) = 1 | + 2·5-s − 10·11-s + 8·19-s − 25-s − 8·29-s − 22·41-s − 11·49-s − 20·55-s + 24·59-s − 14·61-s + 14·71-s + 10·79-s − 6·89-s + 16·95-s + 12·101-s + 24·109-s + 53·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.01·11-s + 1.83·19-s − 1/5·25-s − 1.48·29-s − 3.43·41-s − 1.57·49-s − 2.69·55-s + 3.12·59-s − 1.79·61-s + 1.66·71-s + 1.12·79-s − 0.635·89-s + 1.64·95-s + 1.19·101-s + 2.29·109-s + 4.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6509842487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6509842487\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−8.401311192729721344580035422274, −8.265269534654954141410990237911, −7.53552221236446321492870944747, −7.47015492295014851257914693387, −7.33765788628101262310077533277, −6.48103681477865901897981336596, −6.45410273965163771018212917258, −5.63204685621329517791850004295, −5.57922927235198182161478566706, −5.19159696553250854489100532840, −4.95080524062542653579973794126, −4.74262201340020394324254756137, −3.74406004926410892658320722383, −3.38409052496117103209751156006, −3.23825232365821563417356673251, −2.47766757880110020421314770561, −2.27551006542102180849151253786, −1.80757890890138271899321401355, −1.16649143083525911993073084617, −0.21861776551954550649651597637,
0.21861776551954550649651597637, 1.16649143083525911993073084617, 1.80757890890138271899321401355, 2.27551006542102180849151253786, 2.47766757880110020421314770561, 3.23825232365821563417356673251, 3.38409052496117103209751156006, 3.74406004926410892658320722383, 4.74262201340020394324254756137, 4.95080524062542653579973794126, 5.19159696553250854489100532840, 5.57922927235198182161478566706, 5.63204685621329517791850004295, 6.45410273965163771018212917258, 6.48103681477865901897981336596, 7.33765788628101262310077533277, 7.47015492295014851257914693387, 7.53552221236446321492870944747, 8.265269534654954141410990237911, 8.401311192729721344580035422274
