L(s) = 1 | + (1.39 − 0.244i)2-s + (1.31 + 1.12i)3-s + (1.88 − 0.680i)4-s + (2.10 + 1.25i)6-s − 4.34i·7-s + (2.45 − 1.40i)8-s + (0.448 + 2.96i)9-s − 1.83i·11-s + (3.23 + 1.23i)12-s + 0.588i·13-s + (−1.06 − 6.05i)14-s + (3.07 − 2.55i)16-s + 5.37i·17-s + (1.34 + 4.02i)18-s − 5.38·19-s + ⋯ |
L(s) = 1 | + (0.984 − 0.172i)2-s + (0.758 + 0.652i)3-s + (0.940 − 0.340i)4-s + (0.859 + 0.511i)6-s − 1.64i·7-s + (0.867 − 0.497i)8-s + (0.149 + 0.988i)9-s − 0.553i·11-s + (0.934 + 0.355i)12-s + 0.163i·13-s + (−0.283 − 1.61i)14-s + (0.768 − 0.639i)16-s + 1.30i·17-s + (0.317 + 0.948i)18-s − 1.23·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.36383 - 0.320256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.36383 - 0.320256i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 + 0.244i)T \) |
| 3 | \( 1 + (-1.31 - 1.12i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.34iT - 7T^{2} \) |
| 11 | \( 1 + 1.83iT - 11T^{2} \) |
| 13 | \( 1 - 0.588iT - 13T^{2} \) |
| 17 | \( 1 - 5.37iT - 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 - 7.06iT - 31T^{2} \) |
| 37 | \( 1 - 2.72iT - 37T^{2} \) |
| 41 | \( 1 - 3.42iT - 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 - 9.81T + 47T^{2} \) |
| 53 | \( 1 + 6.65T + 53T^{2} \) |
| 59 | \( 1 - 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 9.27iT - 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 4.42T + 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 + 11.5iT - 83T^{2} \) |
| 89 | \( 1 + 4.21iT - 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
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
−10.66410005853290377620329017757, −10.18036161684282697515977013315, −8.903770126163541897551009135787, −7.86024021617199678527803501434, −7.04729940403364792745155071110, −5.97696505837457167843017454082, −4.63946439858543564947066138709, −3.98752917807885794014094525718, −3.22641115944478796538326408397, −1.67432200816642259972644948148,
2.12656947822912315617696853113, 2.66065144569346782930692920421, 3.99089304078704287993565475874, 5.27317228039981043267767662425, 6.08630056060033112257853977728, 7.06555341140885550059330675974, 7.86486731583879888271772484637, 8.870930134151042666121438246521, 9.545390019377674662568650000114, 11.07137099176468788909632362818