| L(s) = 1 | − 2·2-s − 0.733·3-s + 4·4-s − 5·5-s + 1.46·6-s − 34.7·7-s − 8·8-s − 26.4·9-s + 10·10-s − 11·11-s − 2.93·12-s + 16.8·13-s + 69.5·14-s + 3.66·15-s + 16·16-s + 17·17-s + 52.9·18-s + 36.1·19-s − 20·20-s + 25.4·21-s + 22·22-s + 20.2·23-s + 5.86·24-s + 25·25-s − 33.7·26-s + 39.2·27-s − 139.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.141·3-s + 0.5·4-s − 0.447·5-s + 0.0998·6-s − 1.87·7-s − 0.353·8-s − 0.980·9-s + 0.316·10-s − 0.301·11-s − 0.0705·12-s + 0.359·13-s + 1.32·14-s + 0.0631·15-s + 0.250·16-s + 0.242·17-s + 0.693·18-s + 0.436·19-s − 0.223·20-s + 0.264·21-s + 0.213·22-s + 0.183·23-s + 0.0499·24-s + 0.200·25-s − 0.254·26-s + 0.279·27-s − 0.938·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 0.733T + 27T^{2} \) |
| 7 | \( 1 + 34.7T + 343T^{2} \) |
| 13 | \( 1 - 16.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 36.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 20.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 89.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 395.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 116.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 173.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 363.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 953.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 301.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 645.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 471.T + 9.12e5T^{2} \) |
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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
−8.687106407901957026749130772868, −7.64712629202305245234418945343, −7.04206161120397720804476936356, −6.02983831592130703145352743419, −5.70060267338899172251578793102, −4.12398287384639661020206207275, −3.15445946194705659966577601137, −2.61227224077902794225618062608, −0.841100996309118242017200659272, 0,
0.841100996309118242017200659272, 2.61227224077902794225618062608, 3.15445946194705659966577601137, 4.12398287384639661020206207275, 5.70060267338899172251578793102, 6.02983831592130703145352743419, 7.04206161120397720804476936356, 7.64712629202305245234418945343, 8.687106407901957026749130772868
