| L(s) = 1 | − 8·2-s − 39.6·3-s + 64·4-s − 332.·5-s + 317.·6-s − 135.·7-s − 512·8-s − 614.·9-s + 2.66e3·10-s − 6.18e3·11-s − 2.53e3·12-s − 9.39e3·13-s + 1.08e3·14-s + 1.31e4·15-s + 4.09e3·16-s − 4.30e3·17-s + 4.91e3·18-s − 1.58e4·19-s − 2.12e4·20-s + 5.38e3·21-s + 4.94e4·22-s − 1.04e5·23-s + 2.03e4·24-s + 3.25e4·25-s + 7.51e4·26-s + 1.11e5·27-s − 8.69e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.847·3-s + 0.5·4-s − 1.19·5-s + 0.599·6-s − 0.149·7-s − 0.353·8-s − 0.281·9-s + 0.841·10-s − 1.40·11-s − 0.423·12-s − 1.18·13-s + 0.105·14-s + 1.00·15-s + 0.250·16-s − 0.212·17-s + 0.198·18-s − 0.528·19-s − 0.595·20-s + 0.126·21-s + 0.990·22-s − 1.79·23-s + 0.299·24-s + 0.417·25-s + 0.838·26-s + 1.08·27-s − 0.0748·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1150172936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1150172936\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 37 | \( 1 + 5.06e4T \) |
| good | 3 | \( 1 + 39.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 332.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 135.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.18e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.39e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.30e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.58e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.04e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.39e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.18e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 5.78e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.01e6T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.05e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.22e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.26e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.21e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.89e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.48e6T + 8.07e13T^{2} \) |
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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
−12.62075445162935999074250933097, −11.86852454138445532416450422670, −10.92247183125633917491922270048, −9.969921546821394252877168422631, −8.280346651031260174416178462361, −7.51930271068911929577836760345, −6.04422340256838188417480087770, −4.58670570594933045740155239058, −2.64778034547260045751776077716, −0.24193973982999600292923148944,
0.24193973982999600292923148944, 2.64778034547260045751776077716, 4.58670570594933045740155239058, 6.04422340256838188417480087770, 7.51930271068911929577836760345, 8.280346651031260174416178462361, 9.969921546821394252877168422631, 10.92247183125633917491922270048, 11.86852454138445532416450422670, 12.62075445162935999074250933097
