| L(s) = 1 | + 1.62·2-s − 2.83·3-s − 5.37·4-s + 8.40·5-s − 4.60·6-s + 9.04·7-s − 21.6·8-s − 18.9·9-s + 13.6·10-s + 11·11-s + 15.2·12-s + 14.6·14-s − 23.8·15-s + 7.80·16-s + 121.·17-s − 30.7·18-s − 74.4·19-s − 45.1·20-s − 25.6·21-s + 17.8·22-s − 159.·23-s + 61.5·24-s − 54.2·25-s + 130.·27-s − 48.6·28-s + 268.·29-s − 38.7·30-s + ⋯ |
| L(s) = 1 | + 0.573·2-s − 0.546·3-s − 0.671·4-s + 0.752·5-s − 0.313·6-s + 0.488·7-s − 0.958·8-s − 0.701·9-s + 0.431·10-s + 0.301·11-s + 0.366·12-s + 0.280·14-s − 0.411·15-s + 0.121·16-s + 1.74·17-s − 0.402·18-s − 0.899·19-s − 0.504·20-s − 0.266·21-s + 0.172·22-s − 1.44·23-s + 0.523·24-s − 0.434·25-s + 0.929·27-s − 0.328·28-s + 1.72·29-s − 0.235·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.62T + 8T^{2} \) |
| 3 | \( 1 + 2.83T + 27T^{2} \) |
| 5 | \( 1 - 8.40T + 125T^{2} \) |
| 7 | \( 1 - 9.04T + 343T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 74.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 30.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 422.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 427.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 128.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 622.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 650.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 891.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 730.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.17e3T + 9.12e5T^{2} \) |
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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
−8.405710737829565374099197729421, −7.906767301198353821749277459968, −6.45720410528979916103987953799, −5.86359241898629674940386849670, −5.37256253346314470078499586460, −4.49580764759355627488403115185, −3.59254765015565116568539001764, −2.50385378720377362481698969254, −1.22141041760726930222070359474, 0,
1.22141041760726930222070359474, 2.50385378720377362481698969254, 3.59254765015565116568539001764, 4.49580764759355627488403115185, 5.37256253346314470078499586460, 5.86359241898629674940386849670, 6.45720410528979916103987953799, 7.906767301198353821749277459968, 8.405710737829565374099197729421
