| L(s) = 1 | − 2.67·2-s + 5.15·4-s + 0.481·5-s − 4.15·7-s − 8.44·8-s − 1.28·10-s + 11-s − 13-s + 11.1·14-s + 12.2·16-s + 1.67·17-s + 4.15·19-s + 2.48·20-s − 2.67·22-s + 5.50·23-s − 4.76·25-s + 2.67·26-s − 21.4·28-s − 2.89·29-s + 6.48·31-s − 15.9·32-s − 4.48·34-s − 2·35-s − 6.70·37-s − 11.1·38-s − 4.06·40-s − 2.15·41-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 2.57·4-s + 0.215·5-s − 1.57·7-s − 2.98·8-s − 0.407·10-s + 0.301·11-s − 0.277·13-s + 2.97·14-s + 3.06·16-s + 0.406·17-s + 0.953·19-s + 0.554·20-s − 0.570·22-s + 1.14·23-s − 0.953·25-s + 0.524·26-s − 4.05·28-s − 0.538·29-s + 1.16·31-s − 2.81·32-s − 0.768·34-s − 0.338·35-s − 1.10·37-s − 1.80·38-s − 0.642·40-s − 0.336·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 5 | \( 1 - 0.481T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 4.15T + 19T^{2} \) |
| 23 | \( 1 - 5.50T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 2.15T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 8.57T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 7.66T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.47T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.324880003254705299005324585770, −8.707542545866265629421779786544, −7.70997830790850286264596035814, −6.94171356817892955323394178653, −6.42338291010021937775140831882, −5.43689102834689290072392932599, −3.47439796444725741155662006903, −2.72643462944030860335043872532, −1.36024594680763522143051156293, 0,
1.36024594680763522143051156293, 2.72643462944030860335043872532, 3.47439796444725741155662006903, 5.43689102834689290072392932599, 6.42338291010021937775140831882, 6.94171356817892955323394178653, 7.70997830790850286264596035814, 8.707542545866265629421779786544, 9.324880003254705299005324585770
