| L(s) = 1 | − 8.47·2-s + 1.99·3-s + 39.8·4-s − 10.3·5-s − 16.8·6-s + 175.·7-s − 66.1·8-s − 239.·9-s + 87.9·10-s − 584.·11-s + 79.2·12-s − 668.·13-s − 1.48e3·14-s − 20.6·15-s − 713.·16-s − 439.·17-s + 2.02e3·18-s + 1.59e3·19-s − 413.·20-s + 348.·21-s + 4.95e3·22-s − 1.87e3·23-s − 131.·24-s − 3.01e3·25-s + 5.66e3·26-s − 959.·27-s + 6.97e3·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.127·3-s + 1.24·4-s − 0.185·5-s − 0.191·6-s + 1.35·7-s − 0.365·8-s − 0.983·9-s + 0.278·10-s − 1.45·11-s + 0.158·12-s − 1.09·13-s − 2.02·14-s − 0.0237·15-s − 0.696·16-s − 0.368·17-s + 1.47·18-s + 1.01·19-s − 0.230·20-s + 0.172·21-s + 2.18·22-s − 0.738·23-s − 0.0466·24-s − 0.965·25-s + 1.64·26-s − 0.253·27-s + 1.68·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 1.36e3T \) |
| good | 2 | \( 1 + 8.47T + 32T^{2} \) |
| 3 | \( 1 - 1.99T + 243T^{2} \) |
| 5 | \( 1 + 10.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 584.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 668.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 439.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.59e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.87e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.76e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 1.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.13e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.26e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.11e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.14e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.19e5T + 8.58e9T^{2} \) |
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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
−15.02732560702204246274817737510, −13.75464022310713224392855063512, −11.76508635147675109549137081541, −10.86066286630306350699668321287, −9.588939378605829818832667682406, −8.145684904248472965429304161066, −7.64754677415056570613600658560, −5.18047566977822567525060939769, −2.19920073146592356759519880610, 0,
2.19920073146592356759519880610, 5.18047566977822567525060939769, 7.64754677415056570613600658560, 8.145684904248472965429304161066, 9.588939378605829818832667682406, 10.86066286630306350699668321287, 11.76508635147675109549137081541, 13.75464022310713224392855063512, 15.02732560702204246274817737510
