| L(s) = 1 | + 51.6·2-s − 243·3-s + 617.·4-s − 1.25e4·6-s − 7.43e4·7-s − 7.38e4·8-s + 5.90e4·9-s + 7.15e5·11-s − 1.49e5·12-s − 1.87e6·13-s − 3.84e6·14-s − 5.07e6·16-s − 8.90e5·17-s + 3.04e6·18-s + 1.87e7·19-s + 1.80e7·21-s + 3.69e7·22-s + 5.63e5·23-s + 1.79e7·24-s − 9.69e7·26-s − 1.43e7·27-s − 4.59e7·28-s + 1.17e8·29-s − 1.96e7·31-s − 1.10e8·32-s − 1.73e8·33-s − 4.59e7·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.301·4-s − 0.658·6-s − 1.67·7-s − 0.797·8-s + 0.333·9-s + 1.33·11-s − 0.173·12-s − 1.40·13-s − 1.90·14-s − 1.21·16-s − 0.152·17-s + 0.380·18-s + 1.73·19-s + 0.965·21-s + 1.52·22-s + 0.0182·23-s + 0.460·24-s − 1.59·26-s − 0.192·27-s − 0.504·28-s + 1.06·29-s − 0.123·31-s − 0.583·32-s − 0.773·33-s − 0.173·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.894157769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894157769\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 51.6T + 2.04e3T^{2} \) |
| 7 | \( 1 + 7.43e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.15e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.87e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.90e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.87e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.63e5T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.17e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.96e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.18e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.30e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 4.05e7T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.59e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.33e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.84e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.48e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.58e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 4.54e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.60e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.01e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.70e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.16e11T + 7.15e21T^{2} \) |
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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
−12.22982288083997307724828164474, −11.81423667936863625947881207391, −9.912549493398410965112041183702, −9.274462609106438124060047279304, −7.03382292092208553466015778437, −6.24757457635675262259273689723, −5.10122291081021633418443756008, −3.85790151846453960544671184293, −2.82984589083576600879334618289, −0.63468406291854998280705550143,
0.63468406291854998280705550143, 2.82984589083576600879334618289, 3.85790151846453960544671184293, 5.10122291081021633418443756008, 6.24757457635675262259273689723, 7.03382292092208553466015778437, 9.274462609106438124060047279304, 9.912549493398410965112041183702, 11.81423667936863625947881207391, 12.22982288083997307724828164474
