| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 41.8·5-s − 36·6-s + 175.·7-s − 64·8-s + 81·9-s + 167.·10-s + 766.·11-s + 144·12-s + 577.·13-s − 703.·14-s − 376.·15-s + 256·16-s + 370.·17-s − 324·18-s − 446.·19-s − 668.·20-s + 1.58e3·21-s − 3.06e3·22-s + 956.·23-s − 576·24-s − 1.37e3·25-s − 2.31e3·26-s + 729·27-s + 2.81e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.747·5-s − 0.408·6-s + 1.35·7-s − 0.353·8-s + 0.333·9-s + 0.528·10-s + 1.91·11-s + 0.288·12-s + 0.948·13-s − 0.959·14-s − 0.431·15-s + 0.250·16-s + 0.310·17-s − 0.235·18-s − 0.283·19-s − 0.373·20-s + 0.783·21-s − 1.35·22-s + 0.377·23-s − 0.204·24-s − 0.440·25-s − 0.670·26-s + 0.192·27-s + 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.375771889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375771889\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 + 41.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 766.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 577.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 370.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 446.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 956.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 450.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.72e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.43e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.95e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.44e5T + 8.58e9T^{2} \) |
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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
−10.77039828406991600629554035430, −9.398724354730032818311840144240, −8.745543263640206041146118283354, −7.980065997796403107746123337607, −7.19965644423870970861594648797, −6.00203976085456952136742380060, −4.35923387468719216979986083704, −3.57738351462937076435320000619, −1.84193995041294320823130051026, −1.00046571064026382839507143001,
1.00046571064026382839507143001, 1.84193995041294320823130051026, 3.57738351462937076435320000619, 4.35923387468719216979986083704, 6.00203976085456952136742380060, 7.19965644423870970861594648797, 7.980065997796403107746123337607, 8.745543263640206041146118283354, 9.398724354730032818311840144240, 10.77039828406991600629554035430
