| L(s) = 1 | − 0.517·2-s + 0.462·3-s − 1.73·4-s + 5-s − 0.239·6-s − 3.59·7-s + 1.93·8-s − 2.78·9-s − 0.517·10-s + 1.28·11-s − 0.800·12-s + 6.76·13-s + 1.86·14-s + 0.462·15-s + 2.46·16-s + 4.45·17-s + 1.44·18-s − 5.41·19-s − 1.73·20-s − 1.66·21-s − 0.665·22-s − 1.01·23-s + 0.892·24-s + 25-s − 3.49·26-s − 2.67·27-s + 6.23·28-s + ⋯ |
| L(s) = 1 | − 0.365·2-s + 0.266·3-s − 0.866·4-s + 0.447·5-s − 0.0976·6-s − 1.35·7-s + 0.682·8-s − 0.928·9-s − 0.163·10-s + 0.387·11-s − 0.231·12-s + 1.87·13-s + 0.497·14-s + 0.119·15-s + 0.616·16-s + 1.08·17-s + 0.339·18-s − 1.24·19-s − 0.387·20-s − 0.362·21-s − 0.141·22-s − 0.210·23-s + 0.182·24-s + 0.200·25-s − 0.685·26-s − 0.514·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.517T + 2T^{2} \) |
| 3 | \( 1 - 0.462T + 3T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 - 4.45T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 31 | \( 1 + 0.848T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.04T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 - 0.463T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 4.26T + 97T^{2} \) |
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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
−9.263243114350462775073301449747, −8.616682700244694086495902374543, −8.110900024579261446597500653596, −6.65383445724348974769355662239, −6.07016778739921082700784935728, −5.20326941946869333578968792539, −3.65967397799112866144646732295, −3.38623790356419019905389808094, −1.61431703019164746350083875107, 0,
1.61431703019164746350083875107, 3.38623790356419019905389808094, 3.65967397799112866144646732295, 5.20326941946869333578968792539, 6.07016778739921082700784935728, 6.65383445724348974769355662239, 8.110900024579261446597500653596, 8.616682700244694086495902374543, 9.263243114350462775073301449747
