| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 6.12·11-s − 12-s + 5.68·13-s − 14-s + 15-s + 16-s − 5·17-s + 18-s + 0.561·19-s − 20-s + 21-s − 6.12·22-s − 6.56·23-s − 24-s + 25-s + 5.68·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.84·11-s − 0.288·12-s + 1.57·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 1.21·17-s + 0.235·18-s + 0.128·19-s − 0.223·20-s + 0.218·21-s − 1.30·22-s − 1.36·23-s − 0.204·24-s + 0.200·25-s + 1.11·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 41 | \( 1 - 8.68T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 1.12T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−9.600516873136888235247570747972, −8.369103931345263489013047968798, −7.73589678570944847890062228996, −6.68809362990095016628647885460, −5.93895383136299238805627461161, −5.17864061466260350396003325547, −4.16743143517120259529648606362, −3.30027531686472465337447397895, −1.99602686798888578561163090851, 0,
1.99602686798888578561163090851, 3.30027531686472465337447397895, 4.16743143517120259529648606362, 5.17864061466260350396003325547, 5.93895383136299238805627461161, 6.68809362990095016628647885460, 7.73589678570944847890062228996, 8.369103931345263489013047968798, 9.600516873136888235247570747972
