| L(s) = 1 | − 1.22·2-s + 3-s − 0.489·4-s − 1.22·6-s − 1.54·7-s + 3.05·8-s + 9-s − 5.30·11-s − 0.489·12-s − 1.18·13-s + 1.89·14-s − 2.77·16-s − 0.202·17-s − 1.22·18-s − 0.763·19-s − 1.54·21-s + 6.52·22-s + 0.949·23-s + 3.05·24-s + 1.45·26-s + 27-s + 0.756·28-s − 1.24·29-s + 5.77·31-s − 2.70·32-s − 5.30·33-s + 0.249·34-s + ⋯ |
| L(s) = 1 | − 0.868·2-s + 0.577·3-s − 0.244·4-s − 0.501·6-s − 0.583·7-s + 1.08·8-s + 0.333·9-s − 1.59·11-s − 0.141·12-s − 0.328·13-s + 0.507·14-s − 0.694·16-s − 0.0492·17-s − 0.289·18-s − 0.175·19-s − 0.337·21-s + 1.39·22-s + 0.198·23-s + 0.624·24-s + 0.285·26-s + 0.192·27-s + 0.143·28-s − 0.230·29-s + 1.03·31-s − 0.477·32-s − 0.923·33-s + 0.0427·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7801361797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7801361797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.22T + 2T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 + 0.202T + 17T^{2} \) |
| 19 | \( 1 + 0.763T + 19T^{2} \) |
| 23 | \( 1 - 0.949T + 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 53 | \( 1 - 5.23T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 5.11T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.386836420828334160357687251828, −8.166746053872483289453318673210, −7.30964346773300259989179948050, −6.66615813714328079439283692952, −5.41427762135035071621994706478, −4.81892955262773651045987925718, −3.80318979346130967974715980338, −2.84797976560672121157633835288, −1.99251490051093599911200084910, −0.56062384928475488285171430315,
0.56062384928475488285171430315, 1.99251490051093599911200084910, 2.84797976560672121157633835288, 3.80318979346130967974715980338, 4.81892955262773651045987925718, 5.41427762135035071621994706478, 6.66615813714328079439283692952, 7.30964346773300259989179948050, 8.166746053872483289453318673210, 8.386836420828334160357687251828
