| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 4·13-s − 2·14-s + 16-s − 17-s − 4·19-s + 4·26-s − 2·28-s − 6·29-s − 4·31-s + 32-s − 34-s − 2·37-s − 4·38-s − 2·43-s − 3·49-s + 4·52-s − 6·53-s − 2·56-s − 6·58-s − 6·59-s − 10·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.784·26-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.328·37-s − 0.648·38-s − 0.304·43-s − 3/7·49-s + 0.554·52-s − 0.824·53-s − 0.267·56-s − 0.787·58-s − 0.781·59-s − 1.28·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.43246136992685871321484875653, −6.47639892330129977578028994016, −6.30132932481294227932348717064, −5.45009026326824047700110845372, −4.66570901878524038219320443934, −3.75659402028115115767413150630, −3.40636174922754289850186220550, −2.35081376009060221604372071007, −1.47632560542753140973906046948, 0,
1.47632560542753140973906046948, 2.35081376009060221604372071007, 3.40636174922754289850186220550, 3.75659402028115115767413150630, 4.66570901878524038219320443934, 5.45009026326824047700110845372, 6.30132932481294227932348717064, 6.47639892330129977578028994016, 7.43246136992685871321484875653
