| L(s) = 1 | − 9·3-s − 55.3·5-s + 89.4·7-s + 81·9-s − 566.·11-s − 152.·13-s + 498.·15-s − 2.06e3·17-s − 611.·19-s − 804.·21-s − 529·23-s − 60.3·25-s − 729·27-s + 2.67e3·29-s − 7.99e3·31-s + 5.09e3·33-s − 4.95e3·35-s + 8.76e3·37-s + 1.37e3·39-s + 4.20e3·41-s − 7.35e3·43-s − 4.48e3·45-s − 8.25e3·47-s − 8.80e3·49-s + 1.85e4·51-s + 6.53e3·53-s + 3.13e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.990·5-s + 0.689·7-s + 0.333·9-s − 1.41·11-s − 0.250·13-s + 0.571·15-s − 1.73·17-s − 0.388·19-s − 0.398·21-s − 0.208·23-s − 0.0193·25-s − 0.192·27-s + 0.589·29-s − 1.49·31-s + 0.814·33-s − 0.683·35-s + 1.05·37-s + 0.144·39-s + 0.390·41-s − 0.606·43-s − 0.330·45-s − 0.544·47-s − 0.524·49-s + 0.998·51-s + 0.319·53-s + 1.39·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1946959686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1946959686\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 55.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 89.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 566.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 152.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 611.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.35e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.67e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.38e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.27e4T + 8.58e9T^{2} \) |
| 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
−9.003683612027414607862636615819, −8.110095905012535400736462268321, −7.59276264552523320869257123390, −6.69538714850580319753574519563, −5.62777489634279114336748241681, −4.70928734820460971985940126879, −4.18602418520551642934519277756, −2.80875412294525049769614263422, −1.74956926046200880594986368994, −0.18699280141814451034656056247,
0.18699280141814451034656056247, 1.74956926046200880594986368994, 2.80875412294525049769614263422, 4.18602418520551642934519277756, 4.70928734820460971985940126879, 5.62777489634279114336748241681, 6.69538714850580319753574519563, 7.59276264552523320869257123390, 8.110095905012535400736462268321, 9.003683612027414607862636615819
