| L(s) = 1 | − 78.6·2-s + 4.13e3·4-s + 1.24e4·5-s − 230.·7-s − 1.64e5·8-s − 9.80e5·10-s + 2.67e4·11-s − 1.35e6·13-s + 1.81e4·14-s + 4.44e6·16-s − 3.19e6·17-s − 9.37e6·19-s + 5.15e7·20-s − 2.10e6·22-s + 1.17e7·23-s + 1.06e8·25-s + 1.06e8·26-s − 9.52e5·28-s − 2.05e7·29-s + 1.69e8·31-s − 1.30e7·32-s + 2.51e8·34-s − 2.87e6·35-s − 6.21e8·37-s + 7.37e8·38-s − 2.04e9·40-s + 9.80e8·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.01·4-s + 1.78·5-s − 0.00517·7-s − 1.77·8-s − 3.09·10-s + 0.0500·11-s − 1.01·13-s + 0.00899·14-s + 1.05·16-s − 0.546·17-s − 0.868·19-s + 3.60·20-s − 0.0869·22-s + 0.379·23-s + 2.18·25-s + 1.75·26-s − 0.0104·28-s − 0.185·29-s + 1.06·31-s − 0.0689·32-s + 0.949·34-s − 0.00923·35-s − 1.47·37-s + 1.50·38-s − 3.16·40-s + 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 78.6T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.24e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 230.T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.67e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.35e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.19e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.37e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.17e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 1.69e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.21e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.80e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.55e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.66e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.24e7T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.94e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 6.43e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.26e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.70e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 8.00e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.92e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 7.07e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.54e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.48e10T + 7.15e21T^{2} \) |
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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
−9.358266852728980094863148980650, −9.076697394222937029926775902199, −7.84972625024409654110568995927, −6.76515368623285602420605216548, −6.10923973392139423097104617971, −4.83378333338329415049172719869, −2.66630902472610194422803512483, −2.06632715134879300406915925751, −1.16839971843568429043572829667, 0,
1.16839971843568429043572829667, 2.06632715134879300406915925751, 2.66630902472610194422803512483, 4.83378333338329415049172719869, 6.10923973392139423097104617971, 6.76515368623285602420605216548, 7.84972625024409654110568995927, 9.076697394222937029926775902199, 9.358266852728980094863148980650
