| L(s) = 1 | − 2.75·2-s + 23.9·3-s − 24.4·4-s − 25·5-s − 66.0·6-s + 85.7·7-s + 155.·8-s + 331.·9-s + 68.9·10-s + 431.·11-s − 584.·12-s + 169·13-s − 236.·14-s − 599.·15-s + 352.·16-s − 438.·17-s − 912.·18-s + 1.61e3·19-s + 610.·20-s + 2.05e3·21-s − 1.18e3·22-s + 2.17e3·23-s + 3.72e3·24-s + 625·25-s − 465.·26-s + 2.11e3·27-s − 2.09e3·28-s + ⋯ |
| L(s) = 1 | − 0.487·2-s + 1.53·3-s − 0.762·4-s − 0.447·5-s − 0.749·6-s + 0.661·7-s + 0.858·8-s + 1.36·9-s + 0.217·10-s + 1.07·11-s − 1.17·12-s + 0.277·13-s − 0.322·14-s − 0.687·15-s + 0.343·16-s − 0.368·17-s − 0.664·18-s + 1.02·19-s + 0.341·20-s + 1.01·21-s − 0.523·22-s + 0.856·23-s + 1.32·24-s + 0.200·25-s − 0.135·26-s + 0.557·27-s − 0.504·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.013447731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.013447731\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 2.75T + 32T^{2} \) |
| 3 | \( 1 - 23.9T + 243T^{2} \) |
| 7 | \( 1 - 85.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 431.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 438.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.61e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.17e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.95e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.15e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.75e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.97e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.42e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.02e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.07e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.41e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.33e5T + 8.58e9T^{2} \) |
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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
−14.10634898878246715999983386441, −13.14335344102083375388512153942, −11.61364087843365467244319298526, −10.02556916422162250388607798312, −8.895536632513856232301896720900, −8.394952105535746802981364620725, −7.21930438897657010688068794210, −4.64829563190216059202619554769, −3.39033522329111310498604656107, −1.34248783356685451830123653619,
1.34248783356685451830123653619, 3.39033522329111310498604656107, 4.64829563190216059202619554769, 7.21930438897657010688068794210, 8.394952105535746802981364620725, 8.895536632513856232301896720900, 10.02556916422162250388607798312, 11.61364087843365467244319298526, 13.14335344102083375388512153942, 14.10634898878246715999983386441
