| L(s) = 1 | − 8.58·2-s + 14.9·3-s + 41.7·4-s − 25·5-s − 128.·6-s − 101.·7-s − 83.9·8-s − 18.0·9-s + 214.·10-s + 121·11-s + 626.·12-s − 1.10e3·13-s + 874.·14-s − 374.·15-s − 615.·16-s + 12.4·17-s + 155.·18-s + 361·19-s − 1.04e3·20-s − 1.52e3·21-s − 1.03e3·22-s + 4.62e3·23-s − 1.25e3·24-s + 625·25-s + 9.50e3·26-s − 3.91e3·27-s − 4.25e3·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 0.962·3-s + 1.30·4-s − 0.447·5-s − 1.46·6-s − 0.785·7-s − 0.464·8-s − 0.0743·9-s + 0.679·10-s + 0.301·11-s + 1.25·12-s − 1.81·13-s + 1.19·14-s − 0.430·15-s − 0.601·16-s + 0.0104·17-s + 0.112·18-s + 0.229·19-s − 0.583·20-s − 0.755·21-s − 0.457·22-s + 1.82·23-s − 0.446·24-s + 0.200·25-s + 2.75·26-s − 1.03·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 8.58T + 32T^{2} \) |
| 3 | \( 1 - 14.9T + 243T^{2} \) |
| 7 | \( 1 + 101.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.10e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 12.4T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.63e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 603.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.80e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.35e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.62e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.11e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.05e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.43e5T + 8.58e9T^{2} \) |
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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
−8.936202807866533018406167131544, −8.144337288140014433019034709580, −7.18474963845538277517981568126, −7.02482285577825087770870951457, −5.41635384828015622598582484078, −4.15031381432207188333502507381, −2.90583301719626174181763170478, −2.38101144512483499032116491894, −0.940327474924677605346032730494, 0,
0.940327474924677605346032730494, 2.38101144512483499032116491894, 2.90583301719626174181763170478, 4.15031381432207188333502507381, 5.41635384828015622598582484078, 7.02482285577825087770870951457, 7.18474963845538277517981568126, 8.144337288140014433019034709580, 8.936202807866533018406167131544
