| L(s) = 1 | + 5.60·2-s + 16.2·3-s − 0.624·4-s + 25·5-s + 91.0·6-s + 52.0·7-s − 182.·8-s + 21.0·9-s + 140.·10-s − 121·11-s − 10.1·12-s + 57.9·13-s + 291.·14-s + 406.·15-s − 1.00e3·16-s − 1.40e3·17-s + 117.·18-s + 361·19-s − 15.6·20-s + 846.·21-s − 677.·22-s + 3.31e3·23-s − 2.96e3·24-s + 625·25-s + 324.·26-s − 3.60e3·27-s − 32.5·28-s + ⋯ |
| L(s) = 1 | + 0.990·2-s + 1.04·3-s − 0.0195·4-s + 0.447·5-s + 1.03·6-s + 0.401·7-s − 1.00·8-s + 0.0864·9-s + 0.442·10-s − 0.301·11-s − 0.0203·12-s + 0.0951·13-s + 0.397·14-s + 0.466·15-s − 0.980·16-s − 1.18·17-s + 0.0856·18-s + 0.229·19-s − 0.00873·20-s + 0.418·21-s − 0.298·22-s + 1.30·23-s − 1.05·24-s + 0.200·25-s + 0.0942·26-s − 0.952·27-s − 0.00784·28-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 | \( 1 - 5.60T + 32T^{2} \) |
| 3 | \( 1 - 16.2T + 243T^{2} \) |
| 7 | \( 1 - 52.0T + 1.68e4T^{2} \) |
| 13 | \( 1 - 57.9T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.40e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 505.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.75e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.89e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.51e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.90e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.64e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.33e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.26e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−8.837224080665176340147944706252, −8.130919163788625568064954074003, −6.97522292003959783152507856482, −6.08126174850235334442233522823, −5.06534424977092451625737444372, −4.47036815930353098944503487105, −3.26496285625453768747918622962, −2.74680954868789996694538274161, −1.62543456040358881647504873421, 0,
1.62543456040358881647504873421, 2.74680954868789996694538274161, 3.26496285625453768747918622962, 4.47036815930353098944503487105, 5.06534424977092451625737444372, 6.08126174850235334442233522823, 6.97522292003959783152507856482, 8.130919163788625568064954074003, 8.837224080665176340147944706252
