L(s) = 1 | + 1.07·2-s + 26.0·3-s − 30.8·4-s + 25·5-s + 28.0·6-s − 166.·7-s − 67.6·8-s + 438.·9-s + 26.8·10-s − 121·11-s − 804.·12-s + 807.·13-s − 178.·14-s + 652.·15-s + 914.·16-s − 1.40e3·17-s + 471.·18-s + 361·19-s − 771.·20-s − 4.33e3·21-s − 130.·22-s + 1.46e3·23-s − 1.76e3·24-s + 625·25-s + 868.·26-s + 5.09e3·27-s + 5.12e3·28-s + ⋯ |
L(s) = 1 | + 0.190·2-s + 1.67·3-s − 0.963·4-s + 0.447·5-s + 0.318·6-s − 1.28·7-s − 0.373·8-s + 1.80·9-s + 0.0850·10-s − 0.301·11-s − 1.61·12-s + 1.32·13-s − 0.243·14-s + 0.748·15-s + 0.892·16-s − 1.17·17-s + 0.342·18-s + 0.229·19-s − 0.431·20-s − 2.14·21-s − 0.0573·22-s + 0.578·23-s − 0.625·24-s + 0.200·25-s + 0.252·26-s + 1.34·27-s + 1.23·28-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 | \( 1 - 1.07T + 32T^{2} \) |
| 3 | \( 1 - 26.0T + 243T^{2} \) |
| 7 | \( 1 + 166.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 807.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.40e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.05e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.04e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.20e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.66e4T + 8.58e9T^{2} \) |
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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
−8.822540961276609074026633779573, −8.367916682986451076511786935291, −7.15236593599880689841132900468, −6.32790911690132993606613138117, −5.20294267092929175154585080860, −3.97761739434608311457906423622, −3.43888150245904437562907071892, −2.64613779482800039171843300090, −1.41530020539456391453516909793, 0,
1.41530020539456391453516909793, 2.64613779482800039171843300090, 3.43888150245904437562907071892, 3.97761739434608311457906423622, 5.20294267092929175154585080860, 6.32790911690132993606613138117, 7.15236593599880689841132900468, 8.367916682986451076511786935291, 8.822540961276609074026633779573