| L(s) = 1 | + 0.410·2-s − 1.14·3-s − 1.83·4-s − 0.467·6-s − 3.98·7-s − 1.57·8-s − 1.69·9-s − 3.63·11-s + 2.08·12-s + 1.90·13-s − 1.63·14-s + 3.01·16-s + 4.18·17-s − 0.697·18-s + 8.03·19-s + 4.53·21-s − 1.48·22-s − 8.37·23-s + 1.79·24-s + 0.780·26-s + 5.35·27-s + 7.29·28-s + 3.10·29-s − 6.10·31-s + 4.38·32-s + 4.14·33-s + 1.71·34-s + ⋯ |
| L(s) = 1 | + 0.290·2-s − 0.658·3-s − 0.915·4-s − 0.191·6-s − 1.50·7-s − 0.555·8-s − 0.566·9-s − 1.09·11-s + 0.603·12-s + 0.527·13-s − 0.436·14-s + 0.754·16-s + 1.01·17-s − 0.164·18-s + 1.84·19-s + 0.990·21-s − 0.317·22-s − 1.74·23-s + 0.365·24-s + 0.153·26-s + 1.03·27-s + 1.37·28-s + 0.575·29-s − 1.09·31-s + 0.774·32-s + 0.720·33-s + 0.294·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.410T + 2T^{2} \) |
| 3 | \( 1 + 1.14T + 3T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 3.63T + 11T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 + 8.37T + 23T^{2} \) |
| 29 | \( 1 - 3.10T + 29T^{2} \) |
| 31 | \( 1 + 6.10T + 31T^{2} \) |
| 37 | \( 1 - 0.430T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 6.13T + 53T^{2} \) |
| 59 | \( 1 - 3.54T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 - 0.222T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 7.03T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.82219531461772500044508819675, −6.88354295475966941951767942062, −5.89820546758184237007072602136, −5.67548068411761491954179653258, −5.10292590821733811233927766327, −3.91606483378933434876330927542, −3.35641279337947823035778183061, −2.66485791303642992132411941626, −0.879022766905297533142312165451, 0,
0.879022766905297533142312165451, 2.66485791303642992132411941626, 3.35641279337947823035778183061, 3.91606483378933434876330927542, 5.10292590821733811233927766327, 5.67548068411761491954179653258, 5.89820546758184237007072602136, 6.88354295475966941951767942062, 7.82219531461772500044508819675
