| L(s) = 1 | + 52.4·3-s + 199.·5-s + 343·7-s + 559.·9-s + 1.21e3·11-s + 6.44e3·13-s + 1.04e4·15-s + 2.67e4·17-s + 1.49e4·19-s + 1.79e4·21-s − 7.35e4·23-s − 3.84e4·25-s − 8.52e4·27-s − 1.06e5·29-s + 6.68e4·31-s + 6.38e4·33-s + 6.83e4·35-s − 4.79e5·37-s + 3.37e5·39-s − 6.44e5·41-s + 1.45e5·43-s + 1.11e5·45-s + 1.01e6·47-s + 1.17e5·49-s + 1.40e6·51-s + 3.42e4·53-s + 2.42e5·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s + 0.712·5-s + 0.377·7-s + 0.255·9-s + 0.276·11-s + 0.814·13-s + 0.798·15-s + 1.32·17-s + 0.498·19-s + 0.423·21-s − 1.25·23-s − 0.492·25-s − 0.834·27-s − 0.810·29-s + 0.402·31-s + 0.309·33-s + 0.269·35-s − 1.55·37-s + 0.912·39-s − 1.46·41-s + 0.278·43-s + 0.182·45-s + 1.42·47-s + 0.142·49-s + 1.48·51-s + 0.0315·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.722836203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722836203\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 3 | \( 1 - 52.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 199.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.42e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.72e7T + 8.07e13T^{2} \) |
| 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
−15.42109138499016762990100591992, −14.14500261212495231940646383532, −13.66650631025782096548867400247, −11.93850083991008694361766843338, −10.17678349432128532339976530842, −8.962201325185025528391773777486, −7.76717219542702270248161217236, −5.76430981346213732480806593773, −3.51915384741544475741790919463, −1.76456067451754075735137880176,
1.76456067451754075735137880176, 3.51915384741544475741790919463, 5.76430981346213732480806593773, 7.76717219542702270248161217236, 8.962201325185025528391773777486, 10.17678349432128532339976530842, 11.93850083991008694361766843338, 13.66650631025782096548867400247, 14.14500261212495231940646383532, 15.42109138499016762990100591992
