| L(s) = 1 | − 10.3·2-s − 3.25·3-s + 75.7·4-s + 33.7·6-s − 50.5·7-s − 454.·8-s − 232.·9-s − 121·11-s − 246.·12-s − 990.·13-s + 525.·14-s + 2.29e3·16-s − 427.·17-s + 2.41e3·18-s + 2.33e3·19-s + 164.·21-s + 1.25e3·22-s − 4.20e3·23-s + 1.47e3·24-s + 1.02e4·26-s + 1.54e3·27-s − 3.83e3·28-s + 1.54e3·29-s − 6.33e3·31-s − 9.26e3·32-s + 393.·33-s + 4.43e3·34-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.208·3-s + 2.36·4-s + 0.383·6-s − 0.390·7-s − 2.51·8-s − 0.956·9-s − 0.301·11-s − 0.494·12-s − 1.62·13-s + 0.716·14-s + 2.24·16-s − 0.358·17-s + 1.75·18-s + 1.48·19-s + 0.0814·21-s + 0.553·22-s − 1.65·23-s + 0.524·24-s + 2.98·26-s + 0.408·27-s − 0.924·28-s + 0.340·29-s − 1.18·31-s − 1.60·32-s + 0.0629·33-s + 0.658·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2140989919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2140989919\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 3 | \( 1 + 3.25T + 243T^{2} \) |
| 7 | \( 1 + 50.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 990.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 427.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.33e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.20e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.49e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.79e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.66e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.93e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.86e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.88e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.98e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e5T + 8.58e9T^{2} \) |
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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
−10.76640571532332951063037904829, −9.837214606078333535755446728915, −9.330687308231510067369370362003, −8.162992714569501733990685766102, −7.47485158610776787015991343584, −6.44884762413589452063930017343, −5.27945466485864042904645586997, −3.06434062027717690144530819571, −1.99392290195506979509547184689, −0.33062775347875548383636817176,
0.33062775347875548383636817176, 1.99392290195506979509547184689, 3.06434062027717690144530819571, 5.27945466485864042904645586997, 6.44884762413589452063930017343, 7.47485158610776787015991343584, 8.162992714569501733990685766102, 9.330687308231510067369370362003, 9.837214606078333535755446728915, 10.76640571532332951063037904829
