| L(s) = 1 | − 4.00·2-s + 6.42·3-s + 8.06·4-s − 10.5·5-s − 25.7·6-s + 7·7-s − 0.243·8-s + 14.3·9-s + 42.2·10-s + 51.8·12-s + 60.3·13-s − 28.0·14-s − 67.8·15-s − 63.5·16-s + 33.3·17-s − 57.4·18-s + 69.2·19-s − 85.0·20-s + 44.9·21-s + 158.·23-s − 1.56·24-s − 13.7·25-s − 241.·26-s − 81.4·27-s + 56.4·28-s − 106.·29-s + 271.·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.23·3-s + 1.00·4-s − 0.943·5-s − 1.75·6-s + 0.377·7-s − 0.0107·8-s + 0.530·9-s + 1.33·10-s + 1.24·12-s + 1.28·13-s − 0.535·14-s − 1.16·15-s − 0.992·16-s + 0.476·17-s − 0.751·18-s + 0.836·19-s − 0.950·20-s + 0.467·21-s + 1.44·23-s − 0.0133·24-s − 0.109·25-s − 1.82·26-s − 0.580·27-s + 0.380·28-s − 0.680·29-s + 1.65·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.380583027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380583027\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.00T + 8T^{2} \) |
| 3 | \( 1 - 6.42T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 13 | \( 1 - 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 69.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 133.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 260.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 559.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 837.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 492.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 402.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 856.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 446.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 971.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 779.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.76e3T + 9.12e5T^{2} \) |
| 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
−9.460725880174385999074603390132, −8.731963651829442138361340192164, −8.403202196944052548347872532992, −7.54724503778494531131185807331, −7.07754582821565292896398255011, −5.42784929046123380180982436608, −3.96945959670939617518896578832, −3.23162880820578030153787443255, −1.86691334685343582803074136337, −0.78529799546358322298581348352,
0.78529799546358322298581348352, 1.86691334685343582803074136337, 3.23162880820578030153787443255, 3.96945959670939617518896578832, 5.42784929046123380180982436608, 7.07754582821565292896398255011, 7.54724503778494531131185807331, 8.403202196944052548347872532992, 8.731963651829442138361340192164, 9.460725880174385999074603390132
