| L(s) = 1 | + 0.185·2-s − 0.587·3-s − 1.96·4-s + 5-s − 0.108·6-s + 7-s − 0.734·8-s − 2.65·9-s + 0.185·10-s + 2.91·11-s + 1.15·12-s − 1.78·13-s + 0.185·14-s − 0.587·15-s + 3.79·16-s − 4.66·17-s − 0.492·18-s + 5.48·19-s − 1.96·20-s − 0.587·21-s + 0.539·22-s − 5.72·23-s + 0.431·24-s + 25-s − 0.330·26-s + 3.32·27-s − 1.96·28-s + ⋯ |
| L(s) = 1 | + 0.131·2-s − 0.339·3-s − 0.982·4-s + 0.447·5-s − 0.0444·6-s + 0.377·7-s − 0.259·8-s − 0.884·9-s + 0.0586·10-s + 0.877·11-s + 0.333·12-s − 0.495·13-s + 0.0495·14-s − 0.151·15-s + 0.948·16-s − 1.13·17-s − 0.115·18-s + 1.25·19-s − 0.439·20-s − 0.128·21-s + 0.115·22-s − 1.19·23-s + 0.0881·24-s + 0.200·25-s − 0.0649·26-s + 0.639·27-s − 0.371·28-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 | \( 1 - 0.185T + 2T^{2} \) |
| 3 | \( 1 + 0.587T + 3T^{2} \) |
| 11 | \( 1 - 2.91T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 - 5.48T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 + 4.83T + 53T^{2} \) |
| 59 | \( 1 + 1.67T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 2.77T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 7.29T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 + 7.33T + 97T^{2} \) |
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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
−7.59472925035787031326217480245, −6.58776221105617484980392894845, −6.01578892614341064040723466642, −5.34336879378036166161697539411, −4.74662843514195314352407700698, −4.05310699699894796546225237866, −3.18925179000588005332641470653, −2.22785929664499992654632309363, −1.12059100862740713278757130165, 0,
1.12059100862740713278757130165, 2.22785929664499992654632309363, 3.18925179000588005332641470653, 4.05310699699894796546225237866, 4.74662843514195314352407700698, 5.34336879378036166161697539411, 6.01578892614341064040723466642, 6.58776221105617484980392894845, 7.59472925035787031326217480245
