| L(s) = 1 | + 4·2-s − 11·3-s + 16·4-s − 25·5-s − 44·6-s + 49·7-s + 64·8-s − 122·9-s − 100·10-s − 267·11-s − 176·12-s − 1.08e3·13-s + 196·14-s + 275·15-s + 256·16-s − 513·17-s − 488·18-s − 802·19-s − 400·20-s − 539·21-s − 1.06e3·22-s − 1.29e3·23-s − 704·24-s + 625·25-s − 4.34e3·26-s + 4.01e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.705·3-s + 1/2·4-s − 0.447·5-s − 0.498·6-s + 0.377·7-s + 0.353·8-s − 0.502·9-s − 0.316·10-s − 0.665·11-s − 0.352·12-s − 1.78·13-s + 0.267·14-s + 0.315·15-s + 1/4·16-s − 0.430·17-s − 0.355·18-s − 0.509·19-s − 0.223·20-s − 0.266·21-s − 0.470·22-s − 0.508·23-s − 0.249·24-s + 1/5·25-s − 1.26·26-s + 1.05·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 11 T + p^{5} T^{2} \) |
| 11 | \( 1 + 267 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1087 T + p^{5} T^{2} \) |
| 17 | \( 1 + 513 T + p^{5} T^{2} \) |
| 19 | \( 1 + 802 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1290 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1779 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2584 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13862 T + p^{5} T^{2} \) |
| 41 | \( 1 + 11904 T + p^{5} T^{2} \) |
| 43 | \( 1 + 598 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17019 T + p^{5} T^{2} \) |
| 53 | \( 1 - 27852 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30912 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1780 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25052 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51984 T + p^{5} T^{2} \) |
| 73 | \( 1 - 47690 T + p^{5} T^{2} \) |
| 79 | \( 1 + 102121 T + p^{5} T^{2} \) |
| 83 | \( 1 + 83676 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32400 T + p^{5} T^{2} \) |
| 97 | \( 1 + 148645 T + p^{5} T^{2} \) |
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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
−12.99719828855372312971155156771, −12.01299511998533687622010518081, −11.26247644964533050705817943924, −10.09037948412878920255178156308, −8.224972691902582676820528287811, −6.97074267821956090598598919501, −5.51099302948056373394525686586, −4.52160768341924032493976031559, −2.55783861976934664316050668074, 0,
2.55783861976934664316050668074, 4.52160768341924032493976031559, 5.51099302948056373394525686586, 6.97074267821956090598598919501, 8.224972691902582676820528287811, 10.09037948412878920255178156308, 11.26247644964533050705817943924, 12.01299511998533687622010518081, 12.99719828855372312971155156771
