| L(s) = 1 | + 80.4·2-s + 243·3-s + 4.41e3·4-s + 1.95e4·6-s − 5.39e4·7-s + 1.90e5·8-s + 5.90e4·9-s + 4.07e5·11-s + 1.07e6·12-s + 1.39e6·13-s − 4.33e6·14-s + 6.27e6·16-s + 5.58e6·17-s + 4.74e6·18-s + 7.86e6·19-s − 1.31e7·21-s + 3.27e7·22-s + 6.07e7·23-s + 4.63e7·24-s + 1.11e8·26-s + 1.43e7·27-s − 2.38e8·28-s + 1.28e8·29-s + 1.16e7·31-s + 1.14e8·32-s + 9.89e7·33-s + 4.48e8·34-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.15·4-s + 1.02·6-s − 1.21·7-s + 2.05·8-s + 0.333·9-s + 0.762·11-s + 1.24·12-s + 1.03·13-s − 2.15·14-s + 1.49·16-s + 0.953·17-s + 0.592·18-s + 0.728·19-s − 0.700·21-s + 1.35·22-s + 1.96·23-s + 1.18·24-s + 1.84·26-s + 0.192·27-s − 2.61·28-s + 1.16·29-s + 0.0729·31-s + 0.602·32-s + 0.440·33-s + 1.69·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(8.184482392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.184482392\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 - 80.4T + 2.04e3T^{2} \) |
| 7 | \( 1 + 5.39e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 4.07e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.39e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.58e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.86e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 6.07e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.28e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.16e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.65e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.25e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.69e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.80e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 9.98e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.77e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 6.22e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 6.07e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.98e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.70e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.16e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.63e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.50e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.83e10T + 7.15e21T^{2} \) |
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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
−12.58229820170107425005150706403, −11.65324120114773890422839623695, −10.26238347331605229624285595550, −8.872016385928301695644457100423, −7.06898447741321207168333197589, −6.28721308563795169730104184790, −4.97696578645723661074028813029, −3.42948007427420766696071784964, −3.18204865424604870852123719265, −1.34570258548647150451284697256,
1.34570258548647150451284697256, 3.18204865424604870852123719265, 3.42948007427420766696071784964, 4.97696578645723661074028813029, 6.28721308563795169730104184790, 7.06898447741321207168333197589, 8.872016385928301695644457100423, 10.26238347331605229624285595550, 11.65324120114773890422839623695, 12.58229820170107425005150706403
