| L(s) = 1 | − 2-s + 1.34·3-s + 4-s + 3.90·5-s − 1.34·6-s − 0.487·7-s − 8-s − 1.19·9-s − 3.90·10-s + 4.04·11-s + 1.34·12-s + 3.53·13-s + 0.487·14-s + 5.24·15-s + 16-s − 6.37·17-s + 1.19·18-s + 0.975·19-s + 3.90·20-s − 0.655·21-s − 4.04·22-s + 0.750·23-s − 1.34·24-s + 10.2·25-s − 3.53·26-s − 5.63·27-s − 0.487·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.776·3-s + 0.5·4-s + 1.74·5-s − 0.548·6-s − 0.184·7-s − 0.353·8-s − 0.397·9-s − 1.23·10-s + 1.22·11-s + 0.388·12-s + 0.981·13-s + 0.130·14-s + 1.35·15-s + 0.250·16-s − 1.54·17-s + 0.281·18-s + 0.223·19-s + 0.873·20-s − 0.143·21-s − 0.863·22-s + 0.156·23-s − 0.274·24-s + 2.05·25-s − 0.693·26-s − 1.08·27-s − 0.0922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.772918389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772918389\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 1.34T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.487T + 7T^{2} \) |
| 11 | \( 1 - 4.04T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 - 0.975T + 19T^{2} \) |
| 23 | \( 1 - 0.750T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 2.98T + 59T^{2} \) |
| 61 | \( 1 + 7.53T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.75T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 1.86T + 89T^{2} \) |
| 97 | \( 1 + 9.63T + 97T^{2} \) |
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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
−10.70647176915355308046833031336, −9.565074878449781993456058801529, −9.009815312926693731781485797971, −8.727348526109066696294612888373, −7.18430664689741418114004597618, −6.30196371455321299080964928772, −5.60533450717780083541617327027, −3.80936701963792566063123451023, −2.48415845756228362038085488963, −1.58602000741028475136762805962,
1.58602000741028475136762805962, 2.48415845756228362038085488963, 3.80936701963792566063123451023, 5.60533450717780083541617327027, 6.30196371455321299080964928772, 7.18430664689741418114004597618, 8.727348526109066696294612888373, 9.009815312926693731781485797971, 9.565074878449781993456058801529, 10.70647176915355308046833031336
