| L(s) = 1 | + 2.68·2-s − 2.90·3-s + 5.22·4-s + 3.09·5-s − 7.82·6-s + 1.70·7-s + 8.66·8-s + 5.46·9-s + 8.31·10-s + 11-s − 15.2·12-s + 4.58·14-s − 9.00·15-s + 12.8·16-s + 1.04·17-s + 14.6·18-s − 0.291·19-s + 16.1·20-s − 4.95·21-s + 2.68·22-s − 7.88·23-s − 25.2·24-s + 4.57·25-s − 7.17·27-s + 8.90·28-s − 0.935·29-s − 24.1·30-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 1.68·3-s + 2.61·4-s + 1.38·5-s − 3.19·6-s + 0.644·7-s + 3.06·8-s + 1.82·9-s + 2.62·10-s + 0.301·11-s − 4.38·12-s + 1.22·14-s − 2.32·15-s + 3.21·16-s + 0.252·17-s + 3.46·18-s − 0.0667·19-s + 3.61·20-s − 1.08·21-s + 0.573·22-s − 1.64·23-s − 5.14·24-s + 0.914·25-s − 1.38·27-s + 1.68·28-s − 0.173·29-s − 4.41·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.843255772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.843255772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 + 0.291T + 19T^{2} \) |
| 23 | \( 1 + 7.88T + 23T^{2} \) |
| 29 | \( 1 + 0.935T + 29T^{2} \) |
| 31 | \( 1 - 1.01T + 31T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 + 4.76T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 - 0.582T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 5.79T + 89T^{2} \) |
| 97 | \( 1 + 0.922T + 97T^{2} \) |
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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
−9.827592375266782604554463777181, −8.121508448975128849023689160899, −6.88764644847844978180949146412, −6.49125098078374360340142930801, −5.64426390390831888795870701639, −5.42272911170412415053760104483, −4.63227471385997061686226447431, −3.75233532098444110936882463897, −2.22389768627036907976256747365, −1.47267932417951160364159962718,
1.47267932417951160364159962718, 2.22389768627036907976256747365, 3.75233532098444110936882463897, 4.63227471385997061686226447431, 5.42272911170412415053760104483, 5.64426390390831888795870701639, 6.49125098078374360340142930801, 6.88764644847844978180949146412, 8.121508448975128849023689160899, 9.827592375266782604554463777181
