| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 12·13-s − 16-s − 4·17-s + 2·20-s + 3·25-s + 12·26-s + 4·29-s − 5·32-s + 4·34-s − 4·37-s − 6·40-s + 12·41-s + 49-s − 3·50-s + 12·52-s − 20·53-s − 4·58-s − 4·61-s + 7·64-s + 24·65-s + 4·68-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 3.32·13-s − 1/4·16-s − 0.970·17-s + 0.447·20-s + 3/5·25-s + 2.35·26-s + 0.742·29-s − 0.883·32-s + 0.685·34-s − 0.657·37-s − 0.948·40-s + 1.87·41-s + 1/7·49-s − 0.424·50-s + 1.66·52-s − 2.74·53-s − 0.525·58-s − 0.512·61-s + 7/8·64-s + 2.97·65-s + 0.485·68-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 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
−7.56082201194072800857852642391, −7.33909618476348158334965551669, −6.63672990273285725363430444819, −6.48451110526696282246961711325, −5.50431348153891227960553247972, −5.00126161435509264687569050922, −4.87484917945516723546545158294, −4.19713905019968308039466648355, −4.17084275561163201389021211529, −3.12312009282700194743147534875, −2.65142207448760602119374969734, −2.17599002568151719086793905114, −1.25247251470293336248760296260, 0, 0,
1.25247251470293336248760296260, 2.17599002568151719086793905114, 2.65142207448760602119374969734, 3.12312009282700194743147534875, 4.17084275561163201389021211529, 4.19713905019968308039466648355, 4.87484917945516723546545158294, 5.00126161435509264687569050922, 5.50431348153891227960553247972, 6.48451110526696282246961711325, 6.63672990273285725363430444819, 7.33909618476348158334965551669, 7.56082201194072800857852642391
