| L(s) = 1 | − 1.87·2-s + 1.53·4-s − 2.18·5-s + 0.532·7-s + 0.879·8-s + 4.10·10-s − 11-s − 5.57·13-s − 14-s − 4.71·16-s + 5.87·17-s − 1.29·19-s − 3.34·20-s + 1.87·22-s + 3.06·23-s − 0.226·25-s + 10.4·26-s + 0.815·28-s − 8.58·29-s + 7.82·31-s + 7.10·32-s − 11.0·34-s − 1.16·35-s − 2.36·37-s + 2.42·38-s − 1.92·40-s + 7.47·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.766·4-s − 0.977·5-s + 0.201·7-s + 0.310·8-s + 1.29·10-s − 0.301·11-s − 1.54·13-s − 0.267·14-s − 1.17·16-s + 1.42·17-s − 0.296·19-s − 0.748·20-s + 0.400·22-s + 0.638·23-s − 0.0453·25-s + 2.05·26-s + 0.154·28-s − 1.59·29-s + 1.40·31-s + 1.25·32-s − 1.89·34-s − 0.196·35-s − 0.389·37-s + 0.393·38-s − 0.303·40-s + 1.16·41-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 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 13 | \( 1 + 5.57T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 1.29T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 + 8.58T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 0.290T + 47T^{2} \) |
| 53 | \( 1 + 3.93T + 53T^{2} \) |
| 59 | \( 1 - 3.69T + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 8.51T + 89T^{2} \) |
| 97 | \( 1 + 0.887T + 97T^{2} \) |
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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
−7.66130511119516384084155018804, −7.33826452223930297839309832608, −6.37375818874451430606669454723, −5.29429493797103367788527910132, −4.70539582806255101120364889962, −3.88108652853994956257272094181, −2.91197951681050960842209323211, −2.02044385821037081658335323843, −0.905736404301024281601315200826, 0,
0.905736404301024281601315200826, 2.02044385821037081658335323843, 2.91197951681050960842209323211, 3.88108652853994956257272094181, 4.70539582806255101120364889962, 5.29429493797103367788527910132, 6.37375818874451430606669454723, 7.33826452223930297839309832608, 7.66130511119516384084155018804
