| L(s) = 1 | − 4.08·2-s + 7.19·3-s + 8.67·4-s + 7.90·5-s − 29.3·6-s − 23.1·7-s − 2.73·8-s + 24.8·9-s − 32.2·10-s + 11·11-s + 62.4·12-s + 94.4·14-s + 56.9·15-s − 58.1·16-s + 100.·17-s − 101.·18-s + 54.0·19-s + 68.5·20-s − 166.·21-s − 44.9·22-s − 51.0·23-s − 19.7·24-s − 62.5·25-s − 15.5·27-s − 200.·28-s − 258.·29-s − 232.·30-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.38·3-s + 1.08·4-s + 0.707·5-s − 2.00·6-s − 1.24·7-s − 0.121·8-s + 0.919·9-s − 1.02·10-s + 0.301·11-s + 1.50·12-s + 1.80·14-s + 0.979·15-s − 0.909·16-s + 1.42·17-s − 1.32·18-s + 0.652·19-s + 0.766·20-s − 1.73·21-s − 0.435·22-s − 0.462·23-s − 0.167·24-s − 0.500·25-s − 0.110·27-s − 1.35·28-s − 1.65·29-s − 1.41·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.08T + 8T^{2} \) |
| 3 | \( 1 - 7.19T + 27T^{2} \) |
| 5 | \( 1 - 7.90T + 125T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 51.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 258.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 153.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 106.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 208.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 320.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 132.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 808.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 645.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 343.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 835.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 88.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 846.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−8.766740082566694819759834279797, −7.67624502301673729791765847787, −7.51540527416265818721628850077, −6.36037433600800294236857089937, −5.52810359216919145932365416264, −3.86113869259944958294020352887, −3.16791786159563863421024790861, −2.18032248044622558939237459488, −1.35595026987319186308735656081, 0,
1.35595026987319186308735656081, 2.18032248044622558939237459488, 3.16791786159563863421024790861, 3.86113869259944958294020352887, 5.52810359216919145932365416264, 6.36037433600800294236857089937, 7.51540527416265818721628850077, 7.67624502301673729791765847787, 8.766740082566694819759834279797
