| L(s) = 1 | + 2-s − 4-s + 5-s − 7-s − 3·8-s − 3·9-s + 10-s − 6·11-s − 2·13-s − 14-s − 16-s − 4·17-s − 3·18-s − 4·19-s − 20-s − 6·22-s + 4·23-s + 25-s − 2·26-s + 28-s − 6·29-s − 4·31-s + 5·32-s − 4·34-s − 35-s + 3·36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s − 1.06·8-s − 9-s + 0.316·10-s − 1.80·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.707·18-s − 0.917·19-s − 0.223·20-s − 1.27·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s − 0.685·34-s − 0.169·35-s + 1/2·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 47 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28241062797772, −13.61753701948088, −13.20287717194255, −12.96793614191327, −12.61666799572379, −11.87613424957190, −11.30374405310105, −10.75325340095837, −10.41935350084435, −9.630131982574995, −9.297042033330812, −8.621112157465704, −8.404665154867261, −7.575588194867980, −7.089374069659316, −6.276137077981259, −5.856579991220994, −5.386294877091778, −4.894432396167149, −4.456060658440068, −3.572316904255565, −3.081763967614432, −2.394705451575756, −2.134302324436168, −0.5765696210544982, 0,
0.5765696210544982, 2.134302324436168, 2.394705451575756, 3.081763967614432, 3.572316904255565, 4.456060658440068, 4.894432396167149, 5.386294877091778, 5.856579991220994, 6.276137077981259, 7.089374069659316, 7.575588194867980, 8.404665154867261, 8.621112157465704, 9.297042033330812, 9.630131982574995, 10.41935350084435, 10.75325340095837, 11.30374405310105, 11.87613424957190, 12.61666799572379, 12.96793614191327, 13.20287717194255, 13.61753701948088, 14.28241062797772
