| L(s) = 1 | + 21.2·2-s + 321.·4-s + 138.·5-s − 91.3·7-s + 4.10e3·8-s + 2.93e3·10-s + 1.33e3·11-s + 5.60e3·13-s − 1.93e3·14-s + 4.58e4·16-s − 1.22e4·17-s + 2.76e4·19-s + 4.45e4·20-s + 2.82e4·22-s + 5.46e4·23-s − 5.89e4·25-s + 1.18e5·26-s − 2.93e4·28-s − 2.09e5·29-s − 5.65e4·31-s + 4.46e5·32-s − 2.58e5·34-s − 1.26e4·35-s + 5.71e5·37-s + 5.85e5·38-s + 5.68e5·40-s − 3.10e5·41-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.51·4-s + 0.495·5-s − 0.100·7-s + 2.83·8-s + 0.928·10-s + 0.301·11-s + 0.707·13-s − 0.188·14-s + 2.79·16-s − 0.602·17-s + 0.923·19-s + 1.24·20-s + 0.565·22-s + 0.937·23-s − 0.754·25-s + 1.32·26-s − 0.252·28-s − 1.59·29-s − 0.341·31-s + 2.40·32-s − 1.12·34-s − 0.0498·35-s + 1.85·37-s + 1.73·38-s + 1.40·40-s − 0.703·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.301316370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.301316370\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 2 | \( 1 - 21.2T + 128T^{2} \) |
| 5 | \( 1 - 138.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 91.3T + 8.23e5T^{2} \) |
| 13 | \( 1 - 5.60e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.22e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.76e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.65e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.71e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.16e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.28e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.72e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.99e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.80e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.52e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.65e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.14e7T + 8.07e13T^{2} \) |
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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
−12.90821733512337890230277062480, −11.62397731662439044886454771490, −10.94461211172441336757389418317, −9.401960964513003021156599004880, −7.51249908506827355736114410602, −6.32922343565299287528140734411, −5.48436268425985486932957619823, −4.20605390387454732225309686820, −3.03569373238273678154993076430, −1.63418915620492500801315075667,
1.63418915620492500801315075667, 3.03569373238273678154993076430, 4.20605390387454732225309686820, 5.48436268425985486932957619823, 6.32922343565299287528140734411, 7.51249908506827355736114410602, 9.401960964513003021156599004880, 10.94461211172441336757389418317, 11.62397731662439044886454771490, 12.90821733512337890230277062480
