| L(s) = 1 | − 8.64·2-s + 10.2·3-s + 42.7·4-s + 92.1·5-s − 88.9·6-s + 2.86·7-s − 93.3·8-s − 137.·9-s − 797.·10-s + 571.·11-s + 440.·12-s + 169·13-s − 24.7·14-s + 948.·15-s − 561.·16-s − 855.·17-s + 1.18e3·18-s − 2.35e3·19-s + 3.94e3·20-s + 29.4·21-s − 4.94e3·22-s + 2.50e3·23-s − 960.·24-s + 5.37e3·25-s − 1.46e3·26-s − 3.91e3·27-s + 122.·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.659·3-s + 1.33·4-s + 1.64·5-s − 1.00·6-s + 0.0220·7-s − 0.515·8-s − 0.564·9-s − 2.52·10-s + 1.42·11-s + 0.882·12-s + 0.277·13-s − 0.0337·14-s + 1.08·15-s − 0.548·16-s − 0.718·17-s + 0.863·18-s − 1.49·19-s + 2.20·20-s + 0.0145·21-s − 2.17·22-s + 0.989·23-s − 0.340·24-s + 1.71·25-s − 0.424·26-s − 1.03·27-s + 0.0295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9240266662\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9240266662\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 8.64T + 32T^{2} \) |
| 3 | \( 1 - 10.2T + 243T^{2} \) |
| 5 | \( 1 - 92.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 2.86T + 1.68e4T^{2} \) |
| 11 | \( 1 - 571.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 855.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 144.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 515.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.75e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.34e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.97e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.11e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.73e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.12e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.80e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.28e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.14e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.24e4T + 8.58e9T^{2} \) |
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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
−18.78632808004585341088647532785, −17.33881633245751868208960812130, −16.99881254617018815196141544414, −14.72974340089338398002793793521, −13.43842315411327882951041609900, −10.94201047970936418269581319864, −9.402071326705819612755137670060, −8.753036729332002384396808079993, −6.49233416537680245637163048603, −1.90911682477042387799092390305,
1.90911682477042387799092390305, 6.49233416537680245637163048603, 8.753036729332002384396808079993, 9.402071326705819612755137670060, 10.94201047970936418269581319864, 13.43842315411327882951041609900, 14.72974340089338398002793793521, 16.99881254617018815196141544414, 17.33881633245751868208960812130, 18.78632808004585341088647532785
