| L(s) = 1 | − i·2-s + i·3-s − 4-s + (−2 − i)5-s + 6-s + 2i·7-s + i·8-s − 9-s + (−1 + 2i)10-s + 2·11-s − i·12-s − 6i·13-s + 2·14-s + (1 − 2i)15-s + 16-s + 2i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408·6-s + 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s + 0.603·11-s − 0.288i·12-s − 1.66i·13-s + 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.612627 - 0.144621i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612627 - 0.144621i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
| 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
−17.01977398561147585569982166153, −15.63832276156840719790648214054, −14.82613947152933352288907316812, −12.96250512409104260660029756949, −11.95451807699588727491650213315, −10.79733812128479603885789976626, −9.291364285307452496699123458412, −8.085457671479813666280641047905, −5.35115232455633761080372105825, −3.56141683746843219707232773454,
4.16710137136039276355392879535, 6.63650189600152426497619539048, 7.48664995322902406921466101924, 9.062236764009944146012759381636, 11.03406984828253276966084497277, 12.26964569530337323975039188135, 13.88854108804486098115315613885, 14.60665231932445477324661417962, 16.14791334000609191876902356433, 16.96826441731702099101614986322
