| L(s) = 1 | + 1.16·2-s + 1.72·3-s − 0.642·4-s − 0.583·5-s + 2.00·6-s − 1.01·7-s − 3.07·8-s − 0.0409·9-s − 0.679·10-s − 4.51·11-s − 1.10·12-s + 13-s − 1.18·14-s − 1.00·15-s − 2.30·16-s + 1.55·17-s − 0.0476·18-s − 8.25·19-s + 0.374·20-s − 1.75·21-s − 5.25·22-s − 4.12·23-s − 5.29·24-s − 4.66·25-s + 1.16·26-s − 5.23·27-s + 0.654·28-s + ⋯ |
| L(s) = 1 | + 0.823·2-s + 0.993·3-s − 0.321·4-s − 0.260·5-s + 0.818·6-s − 0.385·7-s − 1.08·8-s − 0.0136·9-s − 0.214·10-s − 1.35·11-s − 0.318·12-s + 0.277·13-s − 0.317·14-s − 0.258·15-s − 0.575·16-s + 0.376·17-s − 0.0112·18-s − 1.89·19-s + 0.0837·20-s − 0.382·21-s − 1.12·22-s − 0.859·23-s − 1.08·24-s − 0.932·25-s + 0.228·26-s − 1.00·27-s + 0.123·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 0.583T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 8.25T + 19T^{2} \) |
| 23 | \( 1 + 4.12T + 23T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 - 8.33T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 9.49T + 43T^{2} \) |
| 47 | \( 1 + 1.78T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 59 | \( 1 + 2.26T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 + 5.98T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 4.30T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−9.045008318946902066823998671246, −8.231683503052328820719554452073, −8.010258198327570471003817847153, −6.51566860506817236225213343290, −5.82828364879737170510546244921, −4.74680637110704096649835455941, −3.98068991022700414069430052167, −3.05648766289527152137951659941, −2.36316879888808962098834762189, 0,
2.36316879888808962098834762189, 3.05648766289527152137951659941, 3.98068991022700414069430052167, 4.74680637110704096649835455941, 5.82828364879737170510546244921, 6.51566860506817236225213343290, 8.010258198327570471003817847153, 8.231683503052328820719554452073, 9.045008318946902066823998671246
