L(s) = 1 | − 8.44·2-s − 9·3-s + 39.2·4-s − 67.3·5-s + 75.9·6-s − 63.5·7-s − 61.1·8-s + 81·9-s + 568.·10-s + 619.·11-s − 353.·12-s − 508.·13-s + 536.·14-s + 606.·15-s − 739.·16-s − 1.27e3·17-s − 683.·18-s + 1.69e3·19-s − 2.64e3·20-s + 571.·21-s − 5.22e3·22-s − 1.46e3·23-s + 550.·24-s + 1.41e3·25-s + 4.29e3·26-s − 729·27-s − 2.49e3·28-s + ⋯ |
L(s) = 1 | − 1.49·2-s − 0.577·3-s + 1.22·4-s − 1.20·5-s + 0.861·6-s − 0.489·7-s − 0.337·8-s + 0.333·9-s + 1.79·10-s + 1.54·11-s − 0.708·12-s − 0.835·13-s + 0.731·14-s + 0.695·15-s − 0.722·16-s − 1.06·17-s − 0.497·18-s + 1.08·19-s − 1.47·20-s + 0.282·21-s − 2.30·22-s − 0.576·23-s + 0.195·24-s + 0.451·25-s + 1.24·26-s − 0.192·27-s − 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2231618860\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2231618860\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 103 | \( 1 - 1.06e4T \) |
good | 2 | \( 1 + 8.44T + 32T^{2} \) |
| 5 | \( 1 + 67.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 63.5T + 1.68e4T^{2} \) |
| 11 | \( 1 - 619.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 508.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.27e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.69e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 558.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.37e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.71e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.27e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.10e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.70e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.09e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.99e4T + 8.58e9T^{2} \) |
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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
−10.79420323271463138136479840794, −9.713959629213497207284863636547, −9.117549685980072222182757272136, −8.030967468902263806136956165160, −7.15372492989213652796791232777, −6.49172703599695212213256341303, −4.71743789349434403728423610053, −3.56611153839976265637256111288, −1.67067514037490050179241836999, −0.34469157638780704747131572257,
0.34469157638780704747131572257, 1.67067514037490050179241836999, 3.56611153839976265637256111288, 4.71743789349434403728423610053, 6.49172703599695212213256341303, 7.15372492989213652796791232777, 8.030967468902263806136956165160, 9.117549685980072222182757272136, 9.713959629213497207284863636547, 10.79420323271463138136479840794