| L(s) = 1 | + 3.64·5-s − 2·7-s + 3.64·11-s − 5.29·13-s − 7.29·17-s − 5.64·19-s + 23-s + 8.29·25-s + 1.29·29-s − 9.29·31-s − 7.29·35-s − 8.93·37-s − 6·41-s − 5.64·43-s + 6·47-s − 3·49-s − 3.64·53-s + 13.2·55-s + 5.64·61-s − 19.2·65-s − 0.937·67-s − 6·71-s + 3.29·73-s − 7.29·77-s + 12.5·79-s − 8.35·83-s − 26.5·85-s + ⋯ |
| L(s) = 1 | + 1.63·5-s − 0.755·7-s + 1.09·11-s − 1.46·13-s − 1.76·17-s − 1.29·19-s + 0.208·23-s + 1.65·25-s + 0.239·29-s − 1.66·31-s − 1.23·35-s − 1.46·37-s − 0.937·41-s − 0.860·43-s + 0.875·47-s − 0.428·49-s − 0.500·53-s + 1.79·55-s + 0.722·61-s − 2.39·65-s − 0.114·67-s − 0.712·71-s + 0.385·73-s − 0.830·77-s + 1.41·79-s − 0.916·83-s − 2.88·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 5.64T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 3.64T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 0.937T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
| 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
−8.668396215754246675055005650996, −7.13312428964090338884834044388, −6.68991196873160555664829653624, −6.18216757456740641298113550233, −5.23350850982509651054349190581, −4.51730904458844689965659116656, −3.41868011976772141844343433347, −2.24626049600624909362497581254, −1.84643151191015255602108550029, 0,
1.84643151191015255602108550029, 2.24626049600624909362497581254, 3.41868011976772141844343433347, 4.51730904458844689965659116656, 5.23350850982509651054349190581, 6.18216757456740641298113550233, 6.68991196873160555664829653624, 7.13312428964090338884834044388, 8.668396215754246675055005650996
