| L(s) = 1 | + 2.70·2-s + 0.788·3-s + 5.31·4-s − 5-s + 2.13·6-s − 7-s + 8.96·8-s − 2.37·9-s − 2.70·10-s − 0.627·11-s + 4.18·12-s − 3.92·13-s − 2.70·14-s − 0.788·15-s + 13.6·16-s − 6.39·17-s − 6.43·18-s − 7.05·19-s − 5.31·20-s − 0.788·21-s − 1.69·22-s − 1.21·23-s + 7.06·24-s + 25-s − 10.6·26-s − 4.23·27-s − 5.31·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 0.455·3-s + 2.65·4-s − 0.447·5-s + 0.870·6-s − 0.377·7-s + 3.16·8-s − 0.792·9-s − 0.855·10-s − 0.189·11-s + 1.20·12-s − 1.08·13-s − 0.722·14-s − 0.203·15-s + 3.40·16-s − 1.55·17-s − 1.51·18-s − 1.61·19-s − 1.18·20-s − 0.171·21-s − 0.361·22-s − 0.253·23-s + 1.44·24-s + 0.200·25-s − 2.08·26-s − 0.815·27-s − 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.788T + 3T^{2} \) |
| 11 | \( 1 + 0.627T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 6.39T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 2.37T + 31T^{2} \) |
| 37 | \( 1 + 6.35T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.47T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 - 6.36T + 73T^{2} \) |
| 79 | \( 1 - 9.02T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 9.00T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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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
−7.10309680609736416126259131337, −6.59843430334241708690345484632, −6.14750441961776329266637074675, −4.99921753132228990605475383271, −4.79664878840325664142643395255, −3.87115665588250600532426119916, −3.30163715807025891745287203914, −2.39552201930290068985221484194, −2.11715845212447760748447406518, 0,
2.11715845212447760748447406518, 2.39552201930290068985221484194, 3.30163715807025891745287203914, 3.87115665588250600532426119916, 4.79664878840325664142643395255, 4.99921753132228990605475383271, 6.14750441961776329266637074675, 6.59843430334241708690345484632, 7.10309680609736416126259131337
