| L(s) = 1 | + 2.43·2-s − 44.8·3-s − 506.·4-s + 1.83e3·5-s − 109.·6-s + 1.43e3·7-s − 2.48e3·8-s − 1.76e4·9-s + 4.47e3·10-s + 4.95e3·11-s + 2.26e4·12-s − 1.70e5·13-s + 3.49e3·14-s − 8.22e4·15-s + 2.53e5·16-s + 6.41e5·17-s − 4.30e4·18-s + 9.88e5·19-s − 9.29e5·20-s − 6.43e4·21-s + 1.20e4·22-s + 9.33e5·23-s + 1.11e5·24-s + 1.41e6·25-s − 4.15e5·26-s + 1.67e6·27-s − 7.26e5·28-s + ⋯ |
| L(s) = 1 | + 0.107·2-s − 0.319·3-s − 0.988·4-s + 1.31·5-s − 0.0344·6-s + 0.225·7-s − 0.214·8-s − 0.897·9-s + 0.141·10-s + 0.101·11-s + 0.315·12-s − 1.65·13-s + 0.0243·14-s − 0.419·15-s + 0.965·16-s + 1.86·17-s − 0.0967·18-s + 1.73·19-s − 1.29·20-s − 0.0721·21-s + 0.0109·22-s + 0.695·23-s + 0.0684·24-s + 0.726·25-s − 0.178·26-s + 0.606·27-s − 0.223·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.706595601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.706595601\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 2.43T + 512T^{2} \) |
| 3 | \( 1 + 44.8T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.83e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.43e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.95e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.70e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.41e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.88e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.41e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.54e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.83e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 6.53e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.22e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.04e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.33e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.70e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.29e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.96e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.45e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.51e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.75e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.58e8T + 7.60e17T^{2} \) |
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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
−14.25513531229169634025333342209, −12.89910928287581244530772043242, −11.76073909978192817003376027014, −9.936654258572862764848267367530, −9.424055474512646348387967142011, −7.73700250936993407234364286156, −5.71054492800186661188850463973, −5.09226665281669444449122285529, −2.90487724155110293652926373398, −0.918748638523440189590341074853,
0.918748638523440189590341074853, 2.90487724155110293652926373398, 5.09226665281669444449122285529, 5.71054492800186661188850463973, 7.73700250936993407234364286156, 9.424055474512646348387967142011, 9.936654258572862764848267367530, 11.76073909978192817003376027014, 12.89910928287581244530772043242, 14.25513531229169634025333342209
