| L(s) = 1 | + 1.13·2-s − 2.60·3-s − 0.714·4-s − 2.95·6-s + 0.407·7-s − 3.07·8-s + 3.77·9-s + 2·11-s + 1.86·12-s − 0.700·13-s + 0.461·14-s − 2.05·16-s + 1.58·17-s + 4.27·18-s + 4.95·19-s − 1.05·21-s + 2.26·22-s + 1.20·23-s + 8.01·24-s − 0.794·26-s − 2.01·27-s − 0.291·28-s + 5.50·29-s + 8.20·31-s + 3.82·32-s − 5.20·33-s + 1.79·34-s + ⋯ |
| L(s) = 1 | + 0.801·2-s − 1.50·3-s − 0.357·4-s − 1.20·6-s + 0.153·7-s − 1.08·8-s + 1.25·9-s + 0.603·11-s + 0.537·12-s − 0.194·13-s + 0.123·14-s − 0.514·16-s + 0.383·17-s + 1.00·18-s + 1.13·19-s − 0.231·21-s + 0.483·22-s + 0.250·23-s + 1.63·24-s − 0.155·26-s − 0.387·27-s − 0.0549·28-s + 1.02·29-s + 1.47·31-s + 0.675·32-s − 0.906·33-s + 0.307·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.126740914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.126740914\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 + 2.60T + 3T^{2} \) |
| 7 | \( 1 - 0.407T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 0.700T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 5.13T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 9.16T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 + 6.49T + 59T^{2} \) |
| 61 | \( 1 + 9.42T + 61T^{2} \) |
| 67 | \( 1 + 3.08T + 67T^{2} \) |
| 71 | \( 1 - 6.86T + 71T^{2} \) |
| 73 | \( 1 - 0.545T + 73T^{2} \) |
| 79 | \( 1 - 5.48T + 79T^{2} \) |
| 83 | \( 1 - 0.974T + 83T^{2} \) |
| 89 | \( 1 + 2.26T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−10.83217163431297771882113320480, −9.888588232404029532613995275416, −9.054524542542970863415798904892, −7.77422892747386018166893449746, −6.55803444338090113880444719495, −5.93976272376763571864796112267, −5.02548798033089710868825923738, −4.44844740086080234971867279209, −3.13557225334101723193890946713, −0.913110224888650774189629819112,
0.913110224888650774189629819112, 3.13557225334101723193890946713, 4.44844740086080234971867279209, 5.02548798033089710868825923738, 5.93976272376763571864796112267, 6.55803444338090113880444719495, 7.77422892747386018166893449746, 9.054524542542970863415798904892, 9.888588232404029532613995275416, 10.83217163431297771882113320480
