| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 3·14-s + 16-s + 3·17-s − 3·23-s − 5·25-s − 3·28-s − 3·29-s + 4·31-s + 32-s + 3·34-s + 7·37-s − 3·41-s + 3·43-s − 3·46-s − 3·47-s + 2·49-s − 5·50-s − 12·53-s − 3·56-s − 3·58-s − 3·59-s − 12·61-s + 4·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.625·23-s − 25-s − 0.566·28-s − 0.557·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s + 1.15·37-s − 0.468·41-s + 0.457·43-s − 0.442·46-s − 0.437·47-s + 2/7·49-s − 0.707·50-s − 1.64·53-s − 0.400·56-s − 0.393·58-s − 0.390·59-s − 1.53·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68338450568066454313396572517, −6.72304329106653412578238218442, −6.16832147086804076176577878504, −5.66727420696428775610162843695, −4.72323635607938463048713149897, −3.93333784052258171204524556305, −3.27334169281393863732932706170, −2.55853722652165949223693627691, −1.44914307616750443770352230025, 0,
1.44914307616750443770352230025, 2.55853722652165949223693627691, 3.27334169281393863732932706170, 3.93333784052258171204524556305, 4.72323635607938463048713149897, 5.66727420696428775610162843695, 6.16832147086804076176577878504, 6.72304329106653412578238218442, 7.68338450568066454313396572517
