| L(s) = 1 | − 0.938·2-s − 3.28·3-s − 1.12·4-s + 0.685·5-s + 3.07·6-s − 0.676·7-s + 2.92·8-s + 7.75·9-s − 0.642·10-s + 2.90·11-s + 3.67·12-s − 3.50·13-s + 0.634·14-s − 2.24·15-s − 0.504·16-s − 1.96·17-s − 7.27·18-s + 5.08·19-s − 0.767·20-s + 2.22·21-s − 2.72·22-s − 3.09·23-s − 9.59·24-s − 4.53·25-s + 3.28·26-s − 15.6·27-s + 0.758·28-s + ⋯ |
| L(s) = 1 | − 0.663·2-s − 1.89·3-s − 0.560·4-s + 0.306·5-s + 1.25·6-s − 0.255·7-s + 1.03·8-s + 2.58·9-s − 0.203·10-s + 0.876·11-s + 1.06·12-s − 0.972·13-s + 0.169·14-s − 0.580·15-s − 0.126·16-s − 0.477·17-s − 1.71·18-s + 1.16·19-s − 0.171·20-s + 0.484·21-s − 0.581·22-s − 0.645·23-s − 1.95·24-s − 0.906·25-s + 0.644·26-s − 3.00·27-s + 0.143·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4033809250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4033809250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 + 0.938T + 2T^{2} \) |
| 3 | \( 1 + 3.28T + 3T^{2} \) |
| 5 | \( 1 - 0.685T + 5T^{2} \) |
| 7 | \( 1 + 0.676T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 + 3.50T + 13T^{2} \) |
| 17 | \( 1 + 1.96T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 41 | \( 1 - 2.50T + 41T^{2} \) |
| 43 | \( 1 - 2.65T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 - 3.94T + 59T^{2} \) |
| 61 | \( 1 - 8.66T + 61T^{2} \) |
| 67 | \( 1 - 7.09T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 - 6.20T + 79T^{2} \) |
| 83 | \( 1 - 0.618T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.860952953801678142784962181535, −9.143490034912793942694505785351, −7.78676159954630899991767646735, −7.10971256747211202535771212850, −6.24105422014880426103567892930, −5.46604216336371547020337225790, −4.70690165732218286854174879859, −3.89550921373401097575444199603, −1.74991925220048418312067200176, −0.57848353479409868812777353401,
0.57848353479409868812777353401, 1.74991925220048418312067200176, 3.89550921373401097575444199603, 4.70690165732218286854174879859, 5.46604216336371547020337225790, 6.24105422014880426103567892930, 7.10971256747211202535771212850, 7.78676159954630899991767646735, 9.143490034912793942694505785351, 9.860952953801678142784962181535
