| L(s) = 1 | − 0.0888·2-s − 1.99·4-s − 0.674·5-s + 0.687·7-s + 0.354·8-s + 0.0599·10-s − 0.536·11-s − 0.628·13-s − 0.0611·14-s + 3.95·16-s + 5.05·17-s − 0.724·19-s + 1.34·20-s + 0.0477·22-s − 7.36·23-s − 4.54·25-s + 0.0558·26-s − 1.37·28-s − 3.61·29-s + 5.74·31-s − 1.06·32-s − 0.449·34-s − 0.463·35-s − 0.110·37-s + 0.0644·38-s − 0.239·40-s + 8.05·41-s + ⋯ |
| L(s) = 1 | − 0.0628·2-s − 0.996·4-s − 0.301·5-s + 0.260·7-s + 0.125·8-s + 0.0189·10-s − 0.161·11-s − 0.174·13-s − 0.0163·14-s + 0.988·16-s + 1.22·17-s − 0.166·19-s + 0.300·20-s + 0.0101·22-s − 1.53·23-s − 0.909·25-s + 0.0109·26-s − 0.258·28-s − 0.671·29-s + 1.03·31-s − 0.187·32-s − 0.0770·34-s − 0.0784·35-s − 0.0182·37-s + 0.0104·38-s − 0.0378·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0888T + 2T^{2} \) |
| 5 | \( 1 + 0.674T + 5T^{2} \) |
| 7 | \( 1 - 0.687T + 7T^{2} \) |
| 11 | \( 1 + 0.536T + 11T^{2} \) |
| 13 | \( 1 + 0.628T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 + 0.724T + 19T^{2} \) |
| 23 | \( 1 + 7.36T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 + 0.110T + 37T^{2} \) |
| 41 | \( 1 - 8.05T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 1.48T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 9.81T + 59T^{2} \) |
| 61 | \( 1 - 6.30T + 61T^{2} \) |
| 67 | \( 1 + 7.02T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 8.10T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.31T + 97T^{2} \) |
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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
−8.646589247212572716620605486048, −7.80127781680660992673515208020, −7.60486699438747749922991373977, −6.05653520149002002048691592277, −5.57887873886652877559314736872, −4.45594936324884624317943551816, −3.97425013100587109608791313613, −2.84793125428024821000196803163, −1.41130062525952854531830798575, 0,
1.41130062525952854531830798575, 2.84793125428024821000196803163, 3.97425013100587109608791313613, 4.45594936324884624317943551816, 5.57887873886652877559314736872, 6.05653520149002002048691592277, 7.60486699438747749922991373977, 7.80127781680660992673515208020, 8.646589247212572716620605486048
