| L(s) = 1 | + 1.04·2-s − 0.916·4-s + 0.554·5-s + 3.51·7-s − 3.03·8-s + 0.576·10-s − 1.49·11-s − 1.85·13-s + 3.65·14-s − 1.32·16-s + 7.11·17-s − 19-s − 0.507·20-s − 1.55·22-s + 5·23-s − 4.69·25-s − 1.93·26-s − 3.21·28-s + 2.14·29-s − 5.45·31-s + 4.68·32-s + 7.40·34-s + 1.94·35-s + 3.22·37-s − 1.04·38-s − 1.68·40-s − 0.696·41-s + ⋯ |
| L(s) = 1 | + 0.736·2-s − 0.458·4-s + 0.247·5-s + 1.32·7-s − 1.07·8-s + 0.182·10-s − 0.450·11-s − 0.515·13-s + 0.976·14-s − 0.331·16-s + 1.72·17-s − 0.229·19-s − 0.113·20-s − 0.331·22-s + 1.04·23-s − 0.938·25-s − 0.379·26-s − 0.608·28-s + 0.397·29-s − 0.980·31-s + 0.829·32-s + 1.27·34-s + 0.329·35-s + 0.530·37-s − 0.168·38-s − 0.266·40-s − 0.108·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.945008251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.945008251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 5 | \( 1 - 0.554T + 5T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 17 | \( 1 - 7.11T + 17T^{2} \) |
| 23 | \( 1 - 5T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 3.22T + 37T^{2} \) |
| 41 | \( 1 + 0.696T + 41T^{2} \) |
| 43 | \( 1 - 0.324T + 43T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 - 5.46T + 59T^{2} \) |
| 61 | \( 1 + 5.72T + 61T^{2} \) |
| 67 | \( 1 + 0.669T + 67T^{2} \) |
| 71 | \( 1 - 0.157T + 71T^{2} \) |
| 73 | \( 1 - 4.50T + 73T^{2} \) |
| 79 | \( 1 - 9.08T + 79T^{2} \) |
| 83 | \( 1 + 6.03T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−7.83648826934344470019406602871, −7.25638384506857662997676339411, −6.17245464483701056317133106216, −5.38500045882503931740985556234, −5.18074382989479908988469356551, −4.41593818065175156212342965772, −3.62752198350240032988840710740, −2.82020256507900726377173445871, −1.86131482387699954738927578003, −0.77960862551291455321023890091,
0.77960862551291455321023890091, 1.86131482387699954738927578003, 2.82020256507900726377173445871, 3.62752198350240032988840710740, 4.41593818065175156212342965772, 5.18074382989479908988469356551, 5.38500045882503931740985556234, 6.17245464483701056317133106216, 7.25638384506857662997676339411, 7.83648826934344470019406602871
