| L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s + 8·11-s + 2·14-s + 5·16-s − 16·22-s − 16·23-s − 6·25-s − 3·28-s + 4·29-s − 6·32-s − 20·37-s − 8·43-s + 24·44-s + 32·46-s + 49-s + 12·50-s − 12·53-s + 4·56-s − 8·58-s + 7·64-s + 8·67-s − 16·71-s + 40·74-s − 8·77-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 2.41·11-s + 0.534·14-s + 5/4·16-s − 3.41·22-s − 3.33·23-s − 6/5·25-s − 0.566·28-s + 0.742·29-s − 1.06·32-s − 3.28·37-s − 1.21·43-s + 3.61·44-s + 4.71·46-s + 1/7·49-s + 1.69·50-s − 1.64·53-s + 0.534·56-s − 1.05·58-s + 7/8·64-s + 0.977·67-s − 1.89·71-s + 4.64·74-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−9.164662289745378790951914691450, −8.882543700432626086795887126363, −8.337652051623380123011624806396, −7.954464142207805330736993023418, −7.36399840290663473896741338506, −6.58813712157544804919343144385, −6.54855688605176886120203881032, −6.01899647743800981343957937964, −5.32455191890654195761811472253, −4.16943092436514571048096348777, −3.83863171110134241641513411613, −3.17941538941685381741709304651, −1.79365373540820646322667624458, −1.71238689353652455033144003099, 0,
1.71238689353652455033144003099, 1.79365373540820646322667624458, 3.17941538941685381741709304651, 3.83863171110134241641513411613, 4.16943092436514571048096348777, 5.32455191890654195761811472253, 6.01899647743800981343957937964, 6.54855688605176886120203881032, 6.58813712157544804919343144385, 7.36399840290663473896741338506, 7.954464142207805330736993023418, 8.337652051623380123011624806396, 8.882543700432626086795887126363, 9.164662289745378790951914691450
