| L(s) = 1 | − 8·2-s + 62.0·3-s + 64·4-s + 42.8·5-s − 496.·6-s + 819.·7-s − 512·8-s + 1.65e3·9-s − 342.·10-s + 728.·11-s + 3.96e3·12-s + 1.43e4·13-s − 6.55e3·14-s + 2.65e3·15-s + 4.09e3·16-s − 3.48e4·17-s − 1.32e4·18-s + 3.87e4·19-s + 2.74e3·20-s + 5.08e4·21-s − 5.83e3·22-s − 6.14e4·23-s − 3.17e4·24-s − 7.62e4·25-s − 1.14e5·26-s − 3.27e4·27-s + 5.24e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.153·5-s − 0.937·6-s + 0.902·7-s − 0.353·8-s + 0.758·9-s − 0.108·10-s + 0.165·11-s + 0.663·12-s + 1.80·13-s − 0.638·14-s + 0.203·15-s + 0.250·16-s − 1.72·17-s − 0.536·18-s + 1.29·19-s + 0.0766·20-s + 1.19·21-s − 0.116·22-s − 1.05·23-s − 0.468·24-s − 0.976·25-s − 1.27·26-s − 0.319·27-s + 0.451·28-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
| \(L(4)\) |
\(\approx\) |
\(2.632441148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.632441148\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 62.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 42.8T + 7.81e4T^{2} \) |
| 7 | \( 1 - 819.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 728.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.43e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.48e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.87e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.14e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.97e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.55e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 7.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.43e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.48e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.49e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.18e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.77e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.47e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.02e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.67777415214370083076010595469, −11.79244311522966693660340573043, −10.76471576410348553482768296803, −9.377339910894864726580943638083, −8.517306159320857213464598726570, −7.83122459663564787031766662487, −6.21661975984554573517147321830, −4.10187065682825513668347355537, −2.54499302609837275849153217123, −1.28407571223361304310102957593,
1.28407571223361304310102957593, 2.54499302609837275849153217123, 4.10187065682825513668347355537, 6.21661975984554573517147321830, 7.83122459663564787031766662487, 8.517306159320857213464598726570, 9.377339910894864726580943638083, 10.76471576410348553482768296803, 11.79244311522966693660340573043, 13.67777415214370083076010595469
