| L(s) = 1 | − 1.60·2-s − 3.19·3-s + 0.587·4-s + 5.13·6-s − 4.87·7-s + 2.27·8-s + 7.18·9-s − 1.54·11-s − 1.87·12-s + 6.58·13-s + 7.84·14-s − 4.82·16-s − 3.07·17-s − 11.5·18-s + 2.72·19-s + 15.5·21-s + 2.48·22-s − 7.62·23-s − 7.25·24-s − 10.5·26-s − 13.3·27-s − 2.86·28-s − 4.16·29-s + 9.93·31-s + 3.22·32-s + 4.93·33-s + 4.93·34-s + ⋯ |
| L(s) = 1 | − 1.13·2-s − 1.84·3-s + 0.293·4-s + 2.09·6-s − 1.84·7-s + 0.803·8-s + 2.39·9-s − 0.466·11-s − 0.540·12-s + 1.82·13-s + 2.09·14-s − 1.20·16-s − 0.744·17-s − 2.72·18-s + 0.624·19-s + 3.39·21-s + 0.530·22-s − 1.59·23-s − 1.48·24-s − 2.07·26-s − 2.56·27-s − 0.541·28-s − 0.773·29-s + 1.78·31-s + 0.569·32-s + 0.858·33-s + 0.846·34-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 | \( 1 + 1.60T + 2T^{2} \) |
| 3 | \( 1 + 3.19T + 3T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 - 2.72T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 - 7.67T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 2.25T + 53T^{2} \) |
| 59 | \( 1 + 3.80T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 - 0.537T + 67T^{2} \) |
| 71 | \( 1 - 5.63T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 - 0.751T + 79T^{2} \) |
| 83 | \( 1 - 9.33T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 - 3.72T + 97T^{2} \) |
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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
−7.75392807515755130395856178683, −6.73230387200722330627404971579, −6.38171258317365557307056511073, −5.96180423321201922127513792393, −4.96839146036718677057120551490, −4.13465967501862393596367265755, −3.33297103428639657743533522391, −1.77266319232417580363012627320, −0.74247004815821125552866365879, 0,
0.74247004815821125552866365879, 1.77266319232417580363012627320, 3.33297103428639657743533522391, 4.13465967501862393596367265755, 4.96839146036718677057120551490, 5.96180423321201922127513792393, 6.38171258317365557307056511073, 6.73230387200722330627404971579, 7.75392807515755130395856178683
