| L(s) = 1 | − 2-s + 3-s + 4-s − 4.07·5-s − 6-s − 8-s − 2·9-s + 4.07·10-s − 1.94·11-s + 12-s + 4.70·13-s − 4.07·15-s + 16-s + 0.440·17-s + 2·18-s + 4.01·19-s − 4.07·20-s + 1.94·22-s + 4.39·23-s − 24-s + 11.6·25-s − 4.70·26-s − 5·27-s − 4.37·29-s + 4.07·30-s − 6.06·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.82·5-s − 0.408·6-s − 0.353·8-s − 0.666·9-s + 1.29·10-s − 0.585·11-s + 0.288·12-s + 1.30·13-s − 1.05·15-s + 0.250·16-s + 0.106·17-s + 0.471·18-s + 0.921·19-s − 0.912·20-s + 0.413·22-s + 0.917·23-s − 0.204·24-s + 2.32·25-s − 0.921·26-s − 0.962·27-s − 0.813·29-s + 0.744·30-s − 1.08·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 11 | \( 1 + 1.94T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 - 0.440T + 17T^{2} \) |
| 19 | \( 1 - 4.01T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 4.37T + 29T^{2} \) |
| 31 | \( 1 + 6.06T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 - 3.31T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.01T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 + 9.01T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 - 5.01T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.160103379539735377769813300219, −7.52907239420334874165976195616, −7.07701271972350280886175427475, −5.92141505492514526913241079434, −5.08821265465391615974785532357, −3.86432554358289873264599533125, −3.42229400462814987894915282336, −2.63865494252295466645022012966, −1.15285395293485744622326012236, 0,
1.15285395293485744622326012236, 2.63865494252295466645022012966, 3.42229400462814987894915282336, 3.86432554358289873264599533125, 5.08821265465391615974785532357, 5.92141505492514526913241079434, 7.07701271972350280886175427475, 7.52907239420334874165976195616, 8.160103379539735377769813300219
