| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s + 9-s − 10-s − 13-s + 16-s + 10·17-s − 18-s + 20-s − 4·25-s + 26-s − 5·29-s − 32-s − 10·34-s + 36-s − 11·37-s − 40-s − 10·41-s + 45-s − 5·49-s + 4·50-s − 52-s − 10·53-s + 5·58-s + 11·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.277·13-s + 1/4·16-s + 2.42·17-s − 0.235·18-s + 0.223·20-s − 4/5·25-s + 0.196·26-s − 0.928·29-s − 0.176·32-s − 1.71·34-s + 1/6·36-s − 1.80·37-s − 0.158·40-s − 1.56·41-s + 0.149·45-s − 5/7·49-s + 0.565·50-s − 0.138·52-s − 1.37·53-s + 0.656·58-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.941240024608645589957210724807, −7.31428256599479068367069784014, −7.11178419796713438551212611582, −6.57242273144600710147726039383, −6.08833027031649805450581993080, −5.50347750620352854708632162019, −5.31842023025557393387272362078, −4.84212174265779990996076905865, −3.93169739491372396993017139924, −3.49134472837876086602143613287, −3.16459458813085296521095826117, −2.30063289439104502240707585530, −1.68352170340396195432400362333, −1.22756936462628483134248846676, 0,
1.22756936462628483134248846676, 1.68352170340396195432400362333, 2.30063289439104502240707585530, 3.16459458813085296521095826117, 3.49134472837876086602143613287, 3.93169739491372396993017139924, 4.84212174265779990996076905865, 5.31842023025557393387272362078, 5.50347750620352854708632162019, 6.08833027031649805450581993080, 6.57242273144600710147726039383, 7.11178419796713438551212611582, 7.31428256599479068367069784014, 7.941240024608645589957210724807
