| L(s) = 1 | + 4.74·2-s + 3·3-s + 14.4·4-s − 4.51·5-s + 14.2·6-s + 30.7·8-s + 9·9-s − 21.4·10-s − 66.8·11-s + 43.4·12-s + 13·13-s − 13.5·15-s + 29.8·16-s − 96.9·17-s + 42.6·18-s − 31.4·19-s − 65.4·20-s − 317.·22-s + 183.·23-s + 92.2·24-s − 104.·25-s + 61.6·26-s + 27·27-s + 112.·29-s − 64.2·30-s + 77.2·31-s − 104.·32-s + ⋯ |
| L(s) = 1 | + 1.67·2-s + 0.577·3-s + 1.81·4-s − 0.403·5-s + 0.967·6-s + 1.35·8-s + 0.333·9-s − 0.677·10-s − 1.83·11-s + 1.04·12-s + 0.277·13-s − 0.233·15-s + 0.467·16-s − 1.38·17-s + 0.558·18-s − 0.380·19-s − 0.731·20-s − 3.07·22-s + 1.66·23-s + 0.784·24-s − 0.836·25-s + 0.464·26-s + 0.192·27-s + 0.718·29-s − 0.391·30-s + 0.447·31-s − 0.575·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 4.74T + 8T^{2} \) |
| 5 | \( 1 + 4.51T + 125T^{2} \) |
| 11 | \( 1 + 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 77.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 54.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 451.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 42.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 530.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 219.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 822.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 872.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 165.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 230.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−8.288347183102860276836602310675, −7.49523734192482130163724293757, −6.72643995566860476874215053249, −5.91051238360503669601829896524, −4.77227033238490162556810899645, −4.62588227147322334931894378046, −3.32870206181933918744786977180, −2.83422529011940116004151051014, −1.90348968547955548894483240166, 0,
1.90348968547955548894483240166, 2.83422529011940116004151051014, 3.32870206181933918744786977180, 4.62588227147322334931894378046, 4.77227033238490162556810899645, 5.91051238360503669601829896524, 6.72643995566860476874215053249, 7.49523734192482130163724293757, 8.288347183102860276836602310675
