| L(s) = 1 | − 1.22·2-s − 0.804·3-s − 0.507·4-s + 0.982·6-s − 3.79·7-s + 3.06·8-s − 2.35·9-s − 1.23·11-s + 0.408·12-s + 3.45·13-s + 4.63·14-s − 2.72·16-s − 3.00·17-s + 2.87·18-s + 3.05·21-s + 1.51·22-s − 6.14·23-s − 2.46·24-s − 4.22·26-s + 4.30·27-s + 1.92·28-s + 4.28·29-s − 5.10·31-s − 2.79·32-s + 0.995·33-s + 3.67·34-s + 1.19·36-s + ⋯ |
| L(s) = 1 | − 0.863·2-s − 0.464·3-s − 0.253·4-s + 0.401·6-s − 1.43·7-s + 1.08·8-s − 0.784·9-s − 0.373·11-s + 0.117·12-s + 0.958·13-s + 1.23·14-s − 0.681·16-s − 0.729·17-s + 0.677·18-s + 0.665·21-s + 0.322·22-s − 1.28·23-s − 0.503·24-s − 0.827·26-s + 0.828·27-s + 0.363·28-s + 0.796·29-s − 0.916·31-s − 0.494·32-s + 0.173·33-s + 0.629·34-s + 0.199·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 3 | \( 1 + 0.804T + 3T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 3.45T + 13T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 - 4.28T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 - 9.13T + 37T^{2} \) |
| 41 | \( 1 + 5.33T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 + 3.33T + 53T^{2} \) |
| 59 | \( 1 - 0.817T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 - 2.52T + 83T^{2} \) |
| 89 | \( 1 + 2.69T + 89T^{2} \) |
| 97 | \( 1 - 3.06T + 97T^{2} \) |
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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
−7.55298692865822944741551431400, −6.57543947351838938430300012661, −6.19373378167561593254842208121, −5.51665253944988516208403519326, −4.57818898420334180254879484281, −3.81981087567753519095711009064, −3.03381015373129016745451968493, −2.06022990950622523505751351019, −0.76316965617679435111699264206, 0,
0.76316965617679435111699264206, 2.06022990950622523505751351019, 3.03381015373129016745451968493, 3.81981087567753519095711009064, 4.57818898420334180254879484281, 5.51665253944988516208403519326, 6.19373378167561593254842208121, 6.57543947351838938430300012661, 7.55298692865822944741551431400
