| L(s) = 1 | − 4.09e3·2-s + 1.24e6·3-s + 1.67e7·4-s − 2.44e8·5-s − 5.10e9·6-s + 3.92e10·7-s − 6.87e10·8-s + 7.08e11·9-s + 1.00e12·10-s − 1.50e13·11-s + 2.09e13·12-s − 1.45e14·13-s − 1.60e14·14-s − 3.04e14·15-s + 2.81e14·16-s + 1.30e15·17-s − 2.90e15·18-s − 9.75e15·19-s − 4.09e15·20-s + 4.89e16·21-s + 6.17e16·22-s + 1.00e17·23-s − 8.57e16·24-s + 5.96e16·25-s + 5.95e17·26-s − 1.73e17·27-s + 6.58e17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.35·3-s + 0.5·4-s − 0.447·5-s − 0.958·6-s + 1.07·7-s − 0.353·8-s + 0.835·9-s + 0.316·10-s − 1.44·11-s + 0.677·12-s − 1.73·13-s − 0.758·14-s − 0.605·15-s + 0.250·16-s + 0.542·17-s − 0.590·18-s − 1.01·19-s − 0.223·20-s + 1.45·21-s + 1.02·22-s + 0.958·23-s − 0.479·24-s + 0.199·25-s + 1.22·26-s − 0.222·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4.09e3T \) |
| 5 | \( 1 + 2.44e8T \) |
| good | 3 | \( 1 - 1.24e6T + 8.47e11T^{2} \) |
| 7 | \( 1 - 3.92e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.50e13T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.45e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.30e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 9.75e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.00e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.41e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 3.30e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 6.55e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 9.42e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.20e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 8.86e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 6.79e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.11e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 3.14e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 4.52e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 3.31e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 3.04e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.45e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 8.80e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.79e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 7.02e24T + 4.66e49T^{2} \) |
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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
−14.57160202439533757262151841499, −12.76376272715006114510808486289, −10.98159554784362422401248329837, −9.492537646922989460941385155021, −8.062759499573733997825898143470, −7.58484558598131789183616466707, −4.86923100680976881261401695389, −2.92572635266540744950986274490, −1.94108107346812939052426865663, 0,
1.94108107346812939052426865663, 2.92572635266540744950986274490, 4.86923100680976881261401695389, 7.58484558598131789183616466707, 8.062759499573733997825898143470, 9.492537646922989460941385155021, 10.98159554784362422401248329837, 12.76376272715006114510808486289, 14.57160202439533757262151841499
