| L(s) = 1 | − 2-s + 4-s − 5·7-s − 8-s + 9-s + 5·14-s + 16-s − 4·17-s − 18-s + 7·23-s − 25-s − 5·28-s + 6·31-s − 32-s + 4·34-s + 36-s − 3·41-s − 7·46-s − 4·47-s + 18·49-s + 50-s + 5·56-s − 6·62-s − 5·63-s + 64-s − 4·68-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 1.33·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 1.45·23-s − 1/5·25-s − 0.944·28-s + 1.07·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s − 0.468·41-s − 1.03·46-s − 0.583·47-s + 18/7·49-s + 0.141·50-s + 0.668·56-s − 0.762·62-s − 0.629·63-s + 1/8·64-s − 0.485·68-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−9.162067957211288873085705517495, −8.624732789506572984561399578125, −8.214313207125443374158773884351, −7.41536238519363049699004658256, −7.02775596256391606064195134387, −6.64694708052321196125214714845, −6.27643455004399683749278252945, −5.68803310320935072739149072273, −4.95901010232037838013941263458, −4.26203387791172527048621528462, −3.59536727679011104745417263537, −2.92270952938265248225465452085, −2.49669822522363695671635344301, −1.26192610534287707146603935351, 0,
1.26192610534287707146603935351, 2.49669822522363695671635344301, 2.92270952938265248225465452085, 3.59536727679011104745417263537, 4.26203387791172527048621528462, 4.95901010232037838013941263458, 5.68803310320935072739149072273, 6.27643455004399683749278252945, 6.64694708052321196125214714845, 7.02775596256391606064195134387, 7.41536238519363049699004658256, 8.214313207125443374158773884351, 8.624732789506572984561399578125, 9.162067957211288873085705517495
