| L(s) = 1 | − 2.06·2-s + 0.265·3-s − 3.71·4-s − 5.08·5-s − 0.549·6-s − 27.2·7-s + 24.2·8-s − 26.9·9-s + 10.5·10-s − 32.8·11-s − 0.987·12-s + 77.0·13-s + 56.4·14-s − 1.35·15-s − 20.4·16-s − 43.9·17-s + 55.7·18-s + 141.·19-s + 18.9·20-s − 7.24·21-s + 67.9·22-s − 161.·23-s + 6.43·24-s − 99.1·25-s − 159.·26-s − 14.3·27-s + 101.·28-s + ⋯ |
| L(s) = 1 | − 0.731·2-s + 0.0511·3-s − 0.464·4-s − 0.454·5-s − 0.0373·6-s − 1.47·7-s + 1.07·8-s − 0.997·9-s + 0.332·10-s − 0.900·11-s − 0.0237·12-s + 1.64·13-s + 1.07·14-s − 0.0232·15-s − 0.319·16-s − 0.626·17-s + 0.729·18-s + 1.70·19-s + 0.211·20-s − 0.0752·21-s + 0.658·22-s − 1.46·23-s + 0.0547·24-s − 0.792·25-s − 1.20·26-s − 0.102·27-s + 0.684·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 37T \) |
| good | 2 | \( 1 + 2.06T + 8T^{2} \) |
| 3 | \( 1 - 0.265T + 27T^{2} \) |
| 5 | \( 1 + 5.08T + 125T^{2} \) |
| 7 | \( 1 + 27.2T + 343T^{2} \) |
| 11 | \( 1 + 32.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 64.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 492.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 73.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 763.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 642.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 268.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 729.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 617.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 459.T + 9.12e5T^{2} \) |
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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
−15.85475960078140462525905433484, −13.85411950810456712926025070476, −13.11723904762022552730467929973, −11.42351641201810620196718023965, −10.05818344811448326782624128105, −8.941858707706378091678447767642, −7.78376681508433605650012954279, −5.87351061841605176309591106106, −3.54833599290655723808860104253, 0,
3.54833599290655723808860104253, 5.87351061841605176309591106106, 7.78376681508433605650012954279, 8.941858707706378091678447767642, 10.05818344811448326782624128105, 11.42351641201810620196718023965, 13.11723904762022552730467929973, 13.85411950810456712926025070476, 15.85475960078140462525905433484
