| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 13-s − 2·14-s + 16-s + 2·19-s + 20-s − 8·23-s + 25-s + 26-s + 2·28-s + 6·29-s + 4·31-s − 32-s + 2·35-s − 2·37-s − 2·38-s − 40-s + 6·41-s − 2·43-s + 8·46-s − 3·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.338·35-s − 0.328·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.304·43-s + 1.17·46-s − 3/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.72799688513037, −12.13256641613510, −11.85237310228868, −11.51672668456082, −10.82973217460204, −10.54170403991867, −9.902322882888798, −9.780245065177993, −9.236295910519077, −8.511761901010927, −8.273161086951585, −7.934817158519330, −7.287712156017489, −6.886944162389289, −6.256104889778389, −5.966890958214367, −5.186420298737053, −5.000035795430123, −4.187787204194905, −3.791462155620443, −2.999270982137882, −2.367818454726312, −2.100460428428916, −1.308503365914371, −0.8729243002503930, 0,
0.8729243002503930, 1.308503365914371, 2.100460428428916, 2.367818454726312, 2.999270982137882, 3.791462155620443, 4.187787204194905, 5.000035795430123, 5.186420298737053, 5.966890958214367, 6.256104889778389, 6.886944162389289, 7.287712156017489, 7.934817158519330, 8.273161086951585, 8.511761901010927, 9.236295910519077, 9.780245065177993, 9.902322882888798, 10.54170403991867, 10.82973217460204, 11.51672668456082, 11.85237310228868, 12.13256641613510, 12.72799688513037
