| L(s) = 1 | − 3-s + 2.91·5-s − 4.51·7-s + 9-s + 11-s + 13-s − 2.91·15-s − 2.40·17-s − 2·19-s + 4.51·21-s + 5.32·23-s + 3.51·25-s − 27-s − 0.103·29-s − 2.40·31-s − 33-s − 13.1·35-s + 6·37-s − 39-s − 3.32·41-s − 7.73·43-s + 2.91·45-s − 0.813·47-s + 13.3·49-s + 2.40·51-s + 2·53-s + 2.91·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30·5-s − 1.70·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.753·15-s − 0.583·17-s − 0.458·19-s + 0.984·21-s + 1.11·23-s + 0.702·25-s − 0.192·27-s − 0.0192·29-s − 0.432·31-s − 0.174·33-s − 2.22·35-s + 0.986·37-s − 0.160·39-s − 0.519·41-s − 1.17·43-s + 0.434·45-s − 0.118·47-s + 1.90·49-s + 0.337·51-s + 0.274·53-s + 0.393·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 17 | \( 1 + 2.40T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 0.103T + 29T^{2} \) |
| 31 | \( 1 + 2.40T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 + 0.813T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.59T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.28948746659853331610735902986, −6.59255776820456372048303185628, −6.33539294368305784716414811113, −5.67678624385801802015507007320, −4.93592295416798982300182948857, −3.94361595377639813826068190891, −3.10355457248204587198888148665, −2.29722093956389179427426531485, −1.25194695714280676500320690387, 0,
1.25194695714280676500320690387, 2.29722093956389179427426531485, 3.10355457248204587198888148665, 3.94361595377639813826068190891, 4.93592295416798982300182948857, 5.67678624385801802015507007320, 6.33539294368305784716414811113, 6.59255776820456372048303185628, 7.28948746659853331610735902986
