| L(s) = 1 | − 3.79·2-s + 1.78·3-s + 6.37·4-s − 15.2·5-s − 6.75·6-s + 7.73·7-s + 6.14·8-s − 23.8·9-s + 57.7·10-s + 11·11-s + 11.3·12-s − 29.3·14-s − 27.1·15-s − 74.3·16-s − 52.0·17-s + 90.3·18-s + 23.0·19-s − 97.2·20-s + 13.7·21-s − 41.7·22-s + 90.8·23-s + 10.9·24-s + 107.·25-s − 90.5·27-s + 49.3·28-s − 22.7·29-s + 103.·30-s + ⋯ |
| L(s) = 1 | − 1.34·2-s + 0.343·3-s + 0.797·4-s − 1.36·5-s − 0.459·6-s + 0.417·7-s + 0.271·8-s − 0.882·9-s + 1.82·10-s + 0.301·11-s + 0.273·12-s − 0.559·14-s − 0.467·15-s − 1.16·16-s − 0.742·17-s + 1.18·18-s + 0.278·19-s − 1.08·20-s + 0.143·21-s − 0.404·22-s + 0.823·23-s + 0.0931·24-s + 0.858·25-s − 0.645·27-s + 0.332·28-s − 0.145·29-s + 0.627·30-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 | \( 1 + 3.79T + 8T^{2} \) |
| 3 | \( 1 - 1.78T + 27T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 7 | \( 1 - 7.73T + 343T^{2} \) |
| 17 | \( 1 + 52.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 108.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 140.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 202.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 345.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 544.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 535.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 663.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 86.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 623.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 824.T + 9.12e5T^{2} \) |
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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
−8.540735932827922650486438046810, −7.922974641983841855402324317993, −7.33422282854721324620002203884, −6.50143369142040499458287123711, −5.12512807140549995025586835417, −4.25483551756658186736333162389, −3.30363882427838699491109193867, −2.18464963357459301099581462304, −0.908443889893547174299214141194, 0,
0.908443889893547174299214141194, 2.18464963357459301099581462304, 3.30363882427838699491109193867, 4.25483551756658186736333162389, 5.12512807140549995025586835417, 6.50143369142040499458287123711, 7.33422282854721324620002203884, 7.922974641983841855402324317993, 8.540735932827922650486438046810
