| L(s) = 1 | + 1.53·2-s + 0.347·4-s − 3.22·5-s − 0.652·7-s − 2.53·8-s − 4.94·10-s − 11-s + 2.29·13-s − 0.999·14-s − 4.57·16-s + 2.46·17-s + 6.71·19-s − 1.12·20-s − 1.53·22-s + 0.694·23-s + 5.41·25-s + 3.50·26-s − 0.226·28-s + 7.43·29-s − 1.36·31-s − 1.94·32-s + 3.78·34-s + 2.10·35-s − 4.45·37-s + 10.2·38-s + 8.17·40-s + 0.509·41-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.173·4-s − 1.44·5-s − 0.246·7-s − 0.895·8-s − 1.56·10-s − 0.301·11-s + 0.635·13-s − 0.267·14-s − 1.14·16-s + 0.598·17-s + 1.54·19-s − 0.250·20-s − 0.326·22-s + 0.144·23-s + 1.08·25-s + 0.688·26-s − 0.0428·28-s + 1.38·29-s − 0.245·31-s − 0.343·32-s + 0.648·34-s + 0.355·35-s − 0.732·37-s + 1.66·38-s + 1.29·40-s + 0.0796·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 13 | \( 1 - 2.29T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 6.71T + 19T^{2} \) |
| 23 | \( 1 - 0.694T + 23T^{2} \) |
| 29 | \( 1 - 7.43T + 29T^{2} \) |
| 31 | \( 1 + 1.36T + 31T^{2} \) |
| 37 | \( 1 + 4.45T + 37T^{2} \) |
| 41 | \( 1 - 0.509T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.71T + 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 + 0.758T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 + 5.31T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.44230629365202268046497304688, −6.71646330030759864163281550250, −5.98789759428953597696977576923, −5.11002905626474496433435428784, −4.74479964418595174172065855687, −3.72649515869341158069157749144, −3.44243451567566677982759439142, −2.77238766854859682937749604794, −1.15388445999667008377253091895, 0,
1.15388445999667008377253091895, 2.77238766854859682937749604794, 3.44243451567566677982759439142, 3.72649515869341158069157749144, 4.74479964418595174172065855687, 5.11002905626474496433435428784, 5.98789759428953597696977576923, 6.71646330030759864163281550250, 7.44230629365202268046497304688
