| L(s) = 1 | + 2·2-s + 4·4-s + 7.94·5-s − 16.6·7-s + 8·8-s + 15.8·10-s + 5.90·13-s − 33.2·14-s + 16·16-s + 5.29·17-s − 19.2·19-s + 31.7·20-s + 63.5·23-s − 61.8·25-s + 11.8·26-s − 66.4·28-s − 204.·29-s − 209.·31-s + 32·32-s + 10.5·34-s − 132·35-s + 35.5·37-s − 38.5·38-s + 63.5·40-s + 116.·41-s − 70.3·43-s + 127.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.710·5-s − 0.896·7-s + 0.353·8-s + 0.502·10-s + 0.126·13-s − 0.634·14-s + 0.250·16-s + 0.0755·17-s − 0.232·19-s + 0.355·20-s + 0.576·23-s − 0.494·25-s + 0.0891·26-s − 0.448·28-s − 1.30·29-s − 1.21·31-s + 0.176·32-s + 0.0534·34-s − 0.637·35-s + 0.158·37-s − 0.164·38-s + 0.251·40-s + 0.445·41-s − 0.249·43-s + 0.407·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 7.94T + 125T^{2} \) |
| 7 | \( 1 + 16.6T + 343T^{2} \) |
| 13 | \( 1 - 5.90T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.29T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 209.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 116.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 70.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 199.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 430.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 198.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 686.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 528.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 690.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 253.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 768.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 93.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 572.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−8.295042635959984203279946864459, −7.28048711081393046551614452864, −6.65314390516332054102920331803, −5.81248967696074726590883748644, −5.35069881628645334102214643411, −4.15537075763804919839569757481, −3.40212912185427238408115820122, −2.46650637606046359725522856325, −1.50937131056665588793146679394, 0,
1.50937131056665588793146679394, 2.46650637606046359725522856325, 3.40212912185427238408115820122, 4.15537075763804919839569757481, 5.35069881628645334102214643411, 5.81248967696074726590883748644, 6.65314390516332054102920331803, 7.28048711081393046551614452864, 8.295042635959984203279946864459
