| L(s) = 1 | − 2.84e9·2-s + 5.77e18·4-s + 6.89e20·5-s − 9.80e25·7-s − 9.84e27·8-s − 1.95e30·10-s + 2.74e31·11-s + 2.74e32·13-s + 2.78e35·14-s + 1.46e37·16-s + 2.27e37·17-s − 1.42e39·19-s + 3.97e39·20-s − 7.81e40·22-s − 1.13e41·23-s − 3.86e42·25-s − 7.80e41·26-s − 5.65e44·28-s − 2.20e44·29-s − 7.87e44·31-s − 1.89e46·32-s − 6.46e46·34-s − 6.75e46·35-s + 5.41e47·37-s + 4.05e48·38-s − 6.78e48·40-s − 3.44e48·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.50·4-s + 0.331·5-s − 1.64·7-s − 2.81·8-s − 0.619·10-s + 0.474·11-s + 0.0290·13-s + 3.07·14-s + 2.75·16-s + 0.673·17-s − 1.42·19-s + 0.828·20-s − 0.888·22-s − 0.332·23-s − 0.890·25-s − 0.0543·26-s − 4.11·28-s − 0.550·29-s − 0.256·31-s − 2.35·32-s − 1.25·34-s − 0.544·35-s + 0.800·37-s + 2.65·38-s − 0.931·40-s − 0.222·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(62-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+61/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(31)\) |
\(\approx\) |
\(0.2603786153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2603786153\) |
| \(L(\frac{63}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 2.84e9T + 2.30e18T^{2} \) |
| 5 | \( 1 - 6.89e20T + 4.33e42T^{2} \) |
| 7 | \( 1 + 9.80e25T + 3.55e51T^{2} \) |
| 11 | \( 1 - 2.74e31T + 3.34e63T^{2} \) |
| 13 | \( 1 - 2.74e32T + 8.92e67T^{2} \) |
| 17 | \( 1 - 2.27e37T + 1.14e75T^{2} \) |
| 19 | \( 1 + 1.42e39T + 1.00e78T^{2} \) |
| 23 | \( 1 + 1.13e41T + 1.16e83T^{2} \) |
| 29 | \( 1 + 2.20e44T + 1.60e89T^{2} \) |
| 31 | \( 1 + 7.87e44T + 9.39e90T^{2} \) |
| 37 | \( 1 - 5.41e47T + 4.57e95T^{2} \) |
| 41 | \( 1 + 3.44e48T + 2.39e98T^{2} \) |
| 43 | \( 1 + 2.45e49T + 4.38e99T^{2} \) |
| 47 | \( 1 + 7.88e50T + 9.95e101T^{2} \) |
| 53 | \( 1 + 4.85e52T + 1.51e105T^{2} \) |
| 59 | \( 1 - 1.37e54T + 1.05e108T^{2} \) |
| 61 | \( 1 - 4.54e53T + 8.03e108T^{2} \) |
| 67 | \( 1 + 8.79e55T + 2.45e111T^{2} \) |
| 71 | \( 1 + 3.27e56T + 8.44e112T^{2} \) |
| 73 | \( 1 + 6.07e56T + 4.59e113T^{2} \) |
| 79 | \( 1 - 1.23e58T + 5.69e115T^{2} \) |
| 83 | \( 1 - 6.61e58T + 1.15e117T^{2} \) |
| 89 | \( 1 - 1.38e59T + 8.18e118T^{2} \) |
| 97 | \( 1 + 3.96e60T + 1.55e121T^{2} \) |
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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
−10.50303731238921105908624575134, −9.730611999848389711484026104069, −9.030593928648579933768472815120, −7.80667479422125569010665581654, −6.60830927140378296230590917429, −6.04078967156251272162018359666, −3.62641843527717786226458652892, −2.50592628488824510431913839400, −1.50080985286448486462856422972, −0.28139589338373244445738118792,
0.28139589338373244445738118792, 1.50080985286448486462856422972, 2.50592628488824510431913839400, 3.62641843527717786226458652892, 6.04078967156251272162018359666, 6.60830927140378296230590917429, 7.80667479422125569010665581654, 9.030593928648579933768472815120, 9.730611999848389711484026104069, 10.50303731238921105908624575134
