| L(s) = 1 | + 0.833·2-s + 0.371·3-s − 1.30·4-s + 0.309·6-s − 3.36·7-s − 2.75·8-s − 2.86·9-s + 4.22·11-s − 0.485·12-s + 2.14·13-s − 2.80·14-s + 0.315·16-s − 5.91·17-s − 2.38·18-s − 1.05·19-s − 1.24·21-s + 3.52·22-s + 1.38·23-s − 1.02·24-s + 1.78·26-s − 2.17·27-s + 4.38·28-s − 8.76·29-s − 4.98·31-s + 5.77·32-s + 1.57·33-s − 4.92·34-s + ⋯ |
| L(s) = 1 | + 0.589·2-s + 0.214·3-s − 0.652·4-s + 0.126·6-s − 1.27·7-s − 0.973·8-s − 0.953·9-s + 1.27·11-s − 0.140·12-s + 0.595·13-s − 0.748·14-s + 0.0787·16-s − 1.43·17-s − 0.562·18-s − 0.241·19-s − 0.272·21-s + 0.750·22-s + 0.289·23-s − 0.209·24-s + 0.350·26-s − 0.419·27-s + 0.829·28-s − 1.62·29-s − 0.894·31-s + 1.02·32-s + 0.273·33-s − 0.845·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.168618997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.168618997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 0.833T + 2T^{2} \) |
| 3 | \( 1 - 0.371T + 3T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 - 5.08T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 + 1.64T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 6.66T + 53T^{2} \) |
| 59 | \( 1 - 0.478T + 59T^{2} \) |
| 61 | \( 1 + 7.83T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 1.27T + 79T^{2} \) |
| 83 | \( 1 + 7.12T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 4.38T + 97T^{2} \) |
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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
−8.319785109022349441971916157137, −7.16622088971229541441117370504, −6.40338182915566503867131434478, −6.02773680595165621943348339314, −5.25199738352818214015342907024, −4.19024687416611463707914832381, −3.71076353388846856611291060553, −3.10695832768329684283091006976, −2.06070829909865242016487437047, −0.49243678522908830161266319843,
0.49243678522908830161266319843, 2.06070829909865242016487437047, 3.10695832768329684283091006976, 3.71076353388846856611291060553, 4.19024687416611463707914832381, 5.25199738352818214015342907024, 6.02773680595165621943348339314, 6.40338182915566503867131434478, 7.16622088971229541441117370504, 8.319785109022349441971916157137
