| L(s) = 1 | − 140·5-s + 1.00e3·7-s + 4.00e3·11-s − 2.23e3·13-s − 1.14e3·17-s + 1.40e4·19-s + 8.80e4·23-s − 5.85e4·25-s + 4.13e4·29-s − 1.83e5·31-s − 1.40e5·35-s − 1.37e5·37-s − 1.23e5·41-s + 6.38e5·43-s − 7.76e5·47-s + 1.78e5·49-s + 9.81e5·53-s − 5.60e5·55-s + 5.68e5·59-s + 1.00e6·61-s + 3.13e5·65-s − 1.08e6·67-s + 2.33e6·71-s + 6.03e5·73-s + 4.00e6·77-s + 1.83e6·79-s + 2.28e6·83-s + ⋯ |
| L(s) = 1 | − 0.500·5-s + 1.10·7-s + 0.907·11-s − 0.282·13-s − 0.0564·17-s + 0.468·19-s + 1.50·23-s − 0.749·25-s + 0.314·29-s − 1.10·31-s − 0.552·35-s − 0.446·37-s − 0.280·41-s + 1.22·43-s − 1.09·47-s + 0.216·49-s + 0.905·53-s − 0.454·55-s + 0.360·59-s + 0.566·61-s + 0.141·65-s − 0.440·67-s + 0.775·71-s + 0.181·73-s + 1.00·77-s + 0.418·79-s + 0.438·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.428427595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.428427595\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 140T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.00e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.00e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.14e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.80e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.13e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.83e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.76e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 5.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.33e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.03e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.83e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.61e6T + 8.07e13T^{2} \) |
| 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
−10.82432706901130798376932692005, −9.524590694250559545617594711978, −8.653438579651008604165790091203, −7.68635548140137991018065445323, −6.85790337578474073014632979046, −5.45411394221452247012066227189, −4.50927386289042522759281307115, −3.43338096694657788396734041890, −1.91551604555231821485165217772, −0.800501883024426834475274806838,
0.800501883024426834475274806838, 1.91551604555231821485165217772, 3.43338096694657788396734041890, 4.50927386289042522759281307115, 5.45411394221452247012066227189, 6.85790337578474073014632979046, 7.68635548140137991018065445323, 8.653438579651008604165790091203, 9.524590694250559545617594711978, 10.82432706901130798376932692005
