| L(s) = 1 | − 2·4-s + 5-s − 7-s − 3·11-s + 13-s + 4·16-s − 3·17-s − 4·19-s − 2·20-s − 9·23-s + 25-s + 2·28-s − 6·29-s + 2·31-s − 35-s − 37-s − 3·41-s + 2·43-s + 6·44-s − 6·47-s − 6·49-s − 2·52-s + 9·53-s − 3·55-s − 12·59-s + 5·61-s − 8·64-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.277·13-s + 16-s − 0.727·17-s − 0.917·19-s − 0.447·20-s − 1.87·23-s + 1/5·25-s + 0.377·28-s − 1.11·29-s + 0.359·31-s − 0.169·35-s − 0.164·37-s − 0.468·41-s + 0.304·43-s + 0.904·44-s − 0.875·47-s − 6/7·49-s − 0.277·52-s + 1.23·53-s − 0.404·55-s − 1.56·59-s + 0.640·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−10.14051697604131672798069750399, −9.458524366082460420320492630341, −8.532147401344436226602590366692, −7.83420991105047635513056919982, −6.46026292798114514724274972377, −5.63142717562375152012516639837, −4.60785132555940900382676705301, −3.60408487766035450933813281986, −2.10991322139674442815207973110, 0,
2.10991322139674442815207973110, 3.60408487766035450933813281986, 4.60785132555940900382676705301, 5.63142717562375152012516639837, 6.46026292798114514724274972377, 7.83420991105047635513056919982, 8.532147401344436226602590366692, 9.458524366082460420320492630341, 10.14051697604131672798069750399
