| L(s) = 1 | + 0.683·2-s − 1.53·4-s − 4.01·5-s + 4.16·7-s − 2.41·8-s − 2.74·10-s − 3.82·11-s + 5.33·13-s + 2.84·14-s + 1.41·16-s + 2.26·17-s − 7.40·19-s + 6.15·20-s − 2.61·22-s + 5.63·23-s + 11.1·25-s + 3.64·26-s − 6.39·28-s + 9.41·29-s + 5.75·31-s + 5.79·32-s + 1.54·34-s − 16.7·35-s + 0.273·37-s − 5.05·38-s + 9.69·40-s − 2.41·41-s + ⋯ |
| L(s) = 1 | + 0.483·2-s − 0.766·4-s − 1.79·5-s + 1.57·7-s − 0.853·8-s − 0.867·10-s − 1.15·11-s + 1.47·13-s + 0.761·14-s + 0.354·16-s + 0.549·17-s − 1.69·19-s + 1.37·20-s − 0.556·22-s + 1.17·23-s + 2.22·25-s + 0.714·26-s − 1.20·28-s + 1.74·29-s + 1.03·31-s + 1.02·32-s + 0.265·34-s − 2.82·35-s + 0.0449·37-s − 0.820·38-s + 1.53·40-s − 0.376·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291300401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291300401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.683T + 2T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 7.40T + 19T^{2} \) |
| 23 | \( 1 - 5.63T + 23T^{2} \) |
| 29 | \( 1 - 9.41T + 29T^{2} \) |
| 31 | \( 1 - 5.75T + 31T^{2} \) |
| 37 | \( 1 - 0.273T + 37T^{2} \) |
| 41 | \( 1 + 2.41T + 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 + 0.860T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 1.48T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 3.99T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 - 1.86T + 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−10.46813869356148247503907472279, −8.673610155349470580422671133689, −8.345441650586440473144781595416, −7.948846238130460289391921842148, −6.68199361003004140968767820279, −5.33176412425824212241782134658, −4.59330800181448061675068184063, −4.04442864732014072603170615707, −2.94338992675182858945871791870, −0.869146742556401305402855334754,
0.869146742556401305402855334754, 2.94338992675182858945871791870, 4.04442864732014072603170615707, 4.59330800181448061675068184063, 5.33176412425824212241782134658, 6.68199361003004140968767820279, 7.948846238130460289391921842148, 8.345441650586440473144781595416, 8.673610155349470580422671133689, 10.46813869356148247503907472279
