| L(s) = 1 | − 16·2-s + 52.4·3-s + 256·4-s + 1.83e3·5-s − 839.·6-s − 4.09e3·8-s − 1.69e4·9-s − 2.93e4·10-s + 9.38e3·11-s + 1.34e4·12-s + 1.79e5·13-s + 9.63e4·15-s + 6.55e4·16-s + 9.78e4·17-s + 2.70e5·18-s + 5.62e5·19-s + 4.70e5·20-s − 1.50e5·22-s − 9.78e5·23-s − 2.14e5·24-s + 1.41e6·25-s − 2.86e6·26-s − 1.92e6·27-s − 4.31e6·29-s − 1.54e6·30-s + 7.97e6·31-s − 1.04e6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.373·3-s + 0.5·4-s + 1.31·5-s − 0.264·6-s − 0.353·8-s − 0.860·9-s − 0.928·10-s + 0.193·11-s + 0.186·12-s + 1.73·13-s + 0.491·15-s + 0.250·16-s + 0.284·17-s + 0.608·18-s + 0.990·19-s + 0.656·20-s − 0.136·22-s − 0.729·23-s − 0.132·24-s + 0.726·25-s − 1.22·26-s − 0.695·27-s − 1.13·29-s − 0.347·30-s + 1.55·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.348036212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.348036212\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 52.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.83e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 9.38e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.79e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 9.78e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.62e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.78e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.31e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.71e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.00e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.21e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.80e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.42e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.42e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.42e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.33e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.77e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.41e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.26e9T + 7.60e17T^{2} \) |
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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
−11.88957094057782725371145816238, −10.83696360152232826560446028902, −9.745359920034234528205641648291, −8.945259315550405161503235636837, −7.976399309074454096539369652185, −6.33808280467746762464433647702, −5.61819184868322243389641373546, −3.43001602999000888579882080514, −2.12902590130438814167142233980, −0.985486229266157748087357108733,
0.985486229266157748087357108733, 2.12902590130438814167142233980, 3.43001602999000888579882080514, 5.61819184868322243389641373546, 6.33808280467746762464433647702, 7.976399309074454096539369652185, 8.945259315550405161503235636837, 9.745359920034234528205641648291, 10.83696360152232826560446028902, 11.88957094057782725371145816238
