| L(s) = 1 | + 8.96e10·2-s + 1.38e17·3-s + 5.68e21·4-s + 3.52e24·5-s + 1.24e28·6-s − 7.32e29·7-s + 2.97e32·8-s + 1.17e34·9-s + 3.16e35·10-s − 9.42e36·11-s + 7.89e38·12-s − 1.70e39·13-s − 6.56e40·14-s + 4.90e41·15-s + 1.32e43·16-s + 4.40e43·17-s + 1.05e45·18-s − 2.77e45·19-s + 2.00e46·20-s − 1.01e47·21-s − 8.45e47·22-s − 2.74e48·23-s + 4.13e49·24-s − 2.99e49·25-s − 1.53e50·26-s + 5.93e50·27-s − 4.16e51·28-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 1.60·3-s + 2.40·4-s + 0.542·5-s + 2.95·6-s − 0.730·7-s + 2.59·8-s + 1.56·9-s + 1.00·10-s − 1.01·11-s + 3.85·12-s − 0.487·13-s − 1.34·14-s + 0.869·15-s + 2.38·16-s + 0.918·17-s + 2.89·18-s − 1.11·19-s + 1.30·20-s − 1.17·21-s − 1.86·22-s − 1.25·23-s + 4.15·24-s − 0.705·25-s − 0.899·26-s + 0.911·27-s − 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(72-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+71/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(36)\) |
\(\approx\) |
\(10.04892632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.04892632\) |
| \(L(\frac{73}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 8.96e10T + 2.36e21T^{2} \) |
| 3 | \( 1 - 1.38e17T + 7.50e33T^{2} \) |
| 5 | \( 1 - 3.52e24T + 4.23e49T^{2} \) |
| 7 | \( 1 + 7.32e29T + 1.00e60T^{2} \) |
| 11 | \( 1 + 9.42e36T + 8.68e73T^{2} \) |
| 13 | \( 1 + 1.70e39T + 1.23e79T^{2} \) |
| 17 | \( 1 - 4.40e43T + 2.30e87T^{2} \) |
| 19 | \( 1 + 2.77e45T + 6.18e90T^{2} \) |
| 23 | \( 1 + 2.74e48T + 4.81e96T^{2} \) |
| 29 | \( 1 - 5.21e51T + 6.76e103T^{2} \) |
| 31 | \( 1 - 3.99e52T + 7.70e105T^{2} \) |
| 37 | \( 1 - 7.75e55T + 2.19e111T^{2} \) |
| 41 | \( 1 - 2.31e56T + 3.21e114T^{2} \) |
| 43 | \( 1 - 1.24e58T + 9.46e115T^{2} \) |
| 47 | \( 1 - 1.88e59T + 5.23e118T^{2} \) |
| 53 | \( 1 + 6.17e60T + 2.65e122T^{2} \) |
| 59 | \( 1 - 6.18e62T + 5.37e125T^{2} \) |
| 61 | \( 1 + 2.16e63T + 5.73e126T^{2} \) |
| 67 | \( 1 - 9.22e64T + 4.48e129T^{2} \) |
| 71 | \( 1 + 7.54e65T + 2.75e131T^{2} \) |
| 73 | \( 1 + 1.91e66T + 1.97e132T^{2} \) |
| 79 | \( 1 - 5.68e65T + 5.38e134T^{2} \) |
| 83 | \( 1 - 2.08e68T + 1.79e136T^{2} \) |
| 89 | \( 1 + 2.05e69T + 2.55e138T^{2} \) |
| 97 | \( 1 - 9.59e69T + 1.15e141T^{2} \) |
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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
−15.88113233458256389252577013041, −14.61753820645113999978707909197, −13.59652354869581596082030302399, −12.61207454429182692165996677402, −10.02037501135702911167615627709, −7.73800164188548063023135327336, −5.99354682246809962928840741640, −4.19594247914254030341103452850, −2.89963734039042422857166593637, −2.17928549147365687300460926319,
2.17928549147365687300460926319, 2.89963734039042422857166593637, 4.19594247914254030341103452850, 5.99354682246809962928840741640, 7.73800164188548063023135327336, 10.02037501135702911167615627709, 12.61207454429182692165996677402, 13.59652354869581596082030302399, 14.61753820645113999978707909197, 15.88113233458256389252577013041
