| L(s) = 1 | − 2.52·3-s + 2.61·5-s + 2.36·7-s + 3.38·9-s − 11-s + 0.166·13-s − 6.60·15-s + 0.826·17-s + 7.11·19-s − 5.98·21-s + 5.85·23-s + 1.83·25-s − 0.980·27-s − 1.35·29-s − 0.354·31-s + 2.52·33-s + 6.18·35-s − 0.0344·37-s − 0.420·39-s − 6.20·41-s − 7.08·43-s + 8.85·45-s + 4.35·47-s − 1.39·49-s − 2.08·51-s + 5.86·53-s − 2.61·55-s + ⋯ |
| L(s) = 1 | − 1.45·3-s + 1.16·5-s + 0.894·7-s + 1.12·9-s − 0.301·11-s + 0.0461·13-s − 1.70·15-s + 0.200·17-s + 1.63·19-s − 1.30·21-s + 1.22·23-s + 0.367·25-s − 0.188·27-s − 0.251·29-s − 0.0637·31-s + 0.439·33-s + 1.04·35-s − 0.00566·37-s − 0.0673·39-s − 0.969·41-s − 1.08·43-s + 1.32·45-s + 0.634·47-s − 0.199·49-s − 0.292·51-s + 0.805·53-s − 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.826219261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826219261\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 13 | \( 1 - 0.166T + 13T^{2} \) |
| 17 | \( 1 - 0.826T + 17T^{2} \) |
| 19 | \( 1 - 7.11T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 + 0.354T + 31T^{2} \) |
| 37 | \( 1 + 0.0344T + 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 - 4.35T + 47T^{2} \) |
| 53 | \( 1 - 5.86T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 3.45T + 67T^{2} \) |
| 71 | \( 1 - 6.91T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 9.14T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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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
−8.020718972476712084492354861984, −7.06609061780399585958852584389, −6.65475341500216013949572132649, −5.65248046255442082443319558244, −5.23849851220794451093467832191, −5.03293101570919670914629845157, −3.76460463473859689544631266982, −2.60850590731367849283751169339, −1.57156103122577491453547207636, −0.832906283127492312222640726681,
0.832906283127492312222640726681, 1.57156103122577491453547207636, 2.60850590731367849283751169339, 3.76460463473859689544631266982, 5.03293101570919670914629845157, 5.23849851220794451093467832191, 5.65248046255442082443319558244, 6.65475341500216013949572132649, 7.06609061780399585958852584389, 8.020718972476712084492354861984
