| L(s) = 1 | + 14.7·2-s − 243·3-s − 1.83e3·4-s − 3.12e3·5-s − 3.57e3·6-s + 7.99e4·7-s − 5.71e4·8-s + 5.90e4·9-s − 4.59e4·10-s + 8.05e5·11-s + 4.45e5·12-s − 1.19e6·13-s + 1.17e6·14-s + 7.59e5·15-s + 2.91e6·16-s + 2.63e6·17-s + 8.69e5·18-s + 1.16e7·19-s + 5.72e6·20-s − 1.94e7·21-s + 1.18e7·22-s + 1.84e7·23-s + 1.38e7·24-s + 9.76e6·25-s − 1.76e7·26-s − 1.43e7·27-s − 1.46e8·28-s + ⋯ |
| L(s) = 1 | + 0.325·2-s − 0.577·3-s − 0.894·4-s − 0.447·5-s − 0.187·6-s + 1.79·7-s − 0.616·8-s + 0.333·9-s − 0.145·10-s + 1.50·11-s + 0.516·12-s − 0.894·13-s + 0.584·14-s + 0.258·15-s + 0.693·16-s + 0.449·17-s + 0.108·18-s + 1.07·19-s + 0.399·20-s − 1.03·21-s + 0.490·22-s + 0.596·23-s + 0.355·24-s + 0.199·25-s − 0.290·26-s − 0.192·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.591658597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591658597\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 5 | \( 1 + 3.12e3T \) |
| good | 2 | \( 1 - 14.7T + 2.04e3T^{2} \) |
| 7 | \( 1 - 7.99e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.05e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.19e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.63e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.16e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.84e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.90e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.01e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 8.06e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.26e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.67e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 8.58e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.52e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.35e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.65e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 7.58e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.75e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.22e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.43e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.44e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 2.04e10T + 7.15e21T^{2} \) |
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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
−17.00060518486258215179990234521, −14.88770711314333628657448618600, −14.14379926331944508923509481931, −12.23966612941843115158081587001, −11.31649929805415295241948919255, −9.282956412132620153395400101089, −7.64125751743185303938353441161, −5.31564481096576657272496312398, −4.16645937042802381379698966810, −1.07452014889870494921406952175,
1.07452014889870494921406952175, 4.16645937042802381379698966810, 5.31564481096576657272496312398, 7.64125751743185303938353441161, 9.282956412132620153395400101089, 11.31649929805415295241948919255, 12.23966612941843115158081587001, 14.14379926331944508923509481931, 14.88770711314333628657448618600, 17.00060518486258215179990234521
