| L(s) = 1 | + 2-s + 4-s − 2.59·5-s + 0.164·7-s + 8-s − 2.59·10-s − 0.573·13-s + 0.164·14-s + 16-s − 2.30·19-s − 2.59·20-s − 0.409·23-s + 1.70·25-s − 0.573·26-s + 0.164·28-s + 8.48·29-s + 5.48·31-s + 32-s − 0.425·35-s − 7.19·37-s − 2.30·38-s − 2.59·40-s + 0.284·41-s − 6.28·43-s − 0.409·46-s + 9.96·47-s − 6.97·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.15·5-s + 0.0620·7-s + 0.353·8-s − 0.819·10-s − 0.159·13-s + 0.0438·14-s + 0.250·16-s − 0.529·19-s − 0.579·20-s − 0.0854·23-s + 0.341·25-s − 0.112·26-s + 0.0310·28-s + 1.57·29-s + 0.985·31-s + 0.176·32-s − 0.0718·35-s − 1.18·37-s − 0.374·38-s − 0.409·40-s + 0.0444·41-s − 0.958·43-s − 0.0604·46-s + 1.45·47-s − 0.996·49-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 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 - 0.164T + 7T^{2} \) |
| 13 | \( 1 + 0.573T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + 0.409T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 0.284T + 41T^{2} \) |
| 43 | \( 1 + 6.28T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
| 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
−7.65029955467169881572926305239, −6.83694023156727471695702411732, −6.36008326893575941426588819614, −5.36482980366035268245627472491, −4.61847373896471480069629173883, −4.12002568332519813471765003279, −3.29034849132543647686495227895, −2.58037105339058908401898203251, −1.36376498506794649506545582209, 0,
1.36376498506794649506545582209, 2.58037105339058908401898203251, 3.29034849132543647686495227895, 4.12002568332519813471765003279, 4.61847373896471480069629173883, 5.36482980366035268245627472491, 6.36008326893575941426588819614, 6.83694023156727471695702411732, 7.65029955467169881572926305239
