L(s) = 1 | − 2-s − 4·7-s + 8-s + 3·9-s − 3·11-s − 10·13-s + 4·14-s − 16-s + 2·17-s − 3·18-s + 5·19-s + 3·22-s + 7·23-s + 10·26-s − 8·29-s + 2·31-s − 2·34-s − 37-s − 5·38-s + 6·41-s + 4·43-s − 7·46-s + 7·47-s + 9·49-s − 9·53-s − 4·56-s + 8·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.51·7-s + 0.353·8-s + 9-s − 0.904·11-s − 2.77·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.14·19-s + 0.639·22-s + 1.45·23-s + 1.96·26-s − 1.48·29-s + 0.359·31-s − 0.342·34-s − 0.164·37-s − 0.811·38-s + 0.937·41-s + 0.609·43-s − 1.03·46-s + 1.02·47-s + 9/7·49-s − 1.23·53-s − 0.534·56-s + 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5797878004\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5797878004\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.74212238638542392568587489661, −11.11584165163924703177341758099, −10.58729930652192783056899494311, −10.13474066569265854156429223107, −9.724475603072259579790942875550, −9.510857966204882455824488225416, −9.327750071305260772839371146644, −8.532657919845425121242453096632, −7.72716939365291428698653931590, −7.32382579857964788089426536430, −7.25978192377663568102751808188, −6.79614066558429530228927765220, −5.62233238596070663715367089269, −5.58120439354575535628816104719, −4.65124743374832625105413764831, −4.34548316261812174988840844362, −3.10073157014197502721780883375, −2.96673359541490778232284975198, −1.98642303142159471363102641943, −0.58332894184136697956191633983,
0.58332894184136697956191633983, 1.98642303142159471363102641943, 2.96673359541490778232284975198, 3.10073157014197502721780883375, 4.34548316261812174988840844362, 4.65124743374832625105413764831, 5.58120439354575535628816104719, 5.62233238596070663715367089269, 6.79614066558429530228927765220, 7.25978192377663568102751808188, 7.32382579857964788089426536430, 7.72716939365291428698653931590, 8.532657919845425121242453096632, 9.327750071305260772839371146644, 9.510857966204882455824488225416, 9.724475603072259579790942875550, 10.13474066569265854156429223107, 10.58729930652192783056899494311, 11.11584165163924703177341758099, 11.74212238638542392568587489661