| L(s) = 1 | + 2.52·2-s + 4.38·4-s + 1.93·5-s + 0.113·7-s + 6.01·8-s + 4.88·10-s − 11-s + 5.88·13-s + 0.287·14-s + 6.43·16-s − 1.78·17-s + 5.41·19-s + 8.46·20-s − 2.52·22-s − 4.94·23-s − 1.26·25-s + 14.8·26-s + 0.499·28-s + 8.48·29-s + 0.794·31-s + 4.21·32-s − 4.51·34-s + 0.220·35-s − 5.89·37-s + 13.6·38-s + 11.6·40-s + 6.11·41-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.19·4-s + 0.864·5-s + 0.0430·7-s + 2.12·8-s + 1.54·10-s − 0.301·11-s + 1.63·13-s + 0.0769·14-s + 1.60·16-s − 0.433·17-s + 1.24·19-s + 1.89·20-s − 0.538·22-s − 1.03·23-s − 0.253·25-s + 2.91·26-s + 0.0943·28-s + 1.57·29-s + 0.142·31-s + 0.745·32-s − 0.773·34-s + 0.0372·35-s − 0.969·37-s + 2.22·38-s + 1.83·40-s + 0.955·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.183761741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.183761741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 - 0.113T + 7T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 0.794T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 + 8.60T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 + 5.19T + 59T^{2} \) |
| 67 | \( 1 + 5.05T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 + 6.98T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 - 5.08T + 97T^{2} \) |
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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
−7.87933282715003914536064843573, −6.99263402523659693782000941117, −6.25464907403342701197410039069, −5.89924144616310173963608267466, −5.24606690480975214388867705711, −4.47278576107294800210204300762, −3.70292200514386063961749893386, −3.01697587602995969478601711931, −2.15564705302196836775402521702, −1.29039412236981957141676437990,
1.29039412236981957141676437990, 2.15564705302196836775402521702, 3.01697587602995969478601711931, 3.70292200514386063961749893386, 4.47278576107294800210204300762, 5.24606690480975214388867705711, 5.89924144616310173963608267466, 6.25464907403342701197410039069, 6.99263402523659693782000941117, 7.87933282715003914536064843573
