| L(s) = 1 | + (0.152 − 0.263i)2-s + (−87.8 − 152. i)3-s + (255. + 443. i)4-s + (209. + 362. i)5-s − 53.5·6-s + (−2.61e3 − 4.53e3i)7-s + 311.·8-s + (−5.58e3 + 9.67e3i)9-s + 127.·10-s + 6.01e4·11-s + (4.49e4 − 7.78e4i)12-s + (−6.72e4 − 1.16e5i)13-s − 1.59e3·14-s + (3.67e4 − 6.35e4i)15-s + (−1.31e5 + 2.26e5i)16-s + (−2.56e5 + 4.44e5i)17-s + ⋯ |
| L(s) = 1 | + (0.00673 − 0.0116i)2-s + (−0.626 − 1.08i)3-s + (0.499 + 0.865i)4-s + (0.149 + 0.259i)5-s − 0.0168·6-s + (−0.412 − 0.713i)7-s + 0.0269·8-s + (−0.283 + 0.491i)9-s + 0.00402·10-s + 1.23·11-s + (0.625 − 1.08i)12-s + (−0.652 − 1.13i)13-s − 0.0110·14-s + (0.187 − 0.324i)15-s + (−0.499 + 0.865i)16-s + (−0.745 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0661995 - 0.642028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0661995 - 0.642028i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-6.26e6 - 9.52e6i)T \) |
| good | 2 | \( 1 + (-0.152 + 0.263i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (87.8 + 152. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-209. - 362. i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + (2.61e3 + 4.53e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 - 6.01e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (6.72e4 + 1.16e5i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (2.56e5 - 4.44e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (3.43e5 + 5.95e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + 2.58e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.89e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + (9.49e6 + 1.64e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + 2.46e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.87e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + (-3.45e7 + 5.97e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-4.86e7 + 8.43e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-9.56e7 - 1.65e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (6.27e7 + 1.08e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-6.60e6 - 1.14e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 - 1.35e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + (4.59e7 + 7.95e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.13e8 + 5.42e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (6.69e7 - 1.16e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.48e9T + 7.60e17T^{2} \) |
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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
−13.36660128643065514667833691385, −12.58131462482396344855842669474, −11.65298484035906000098239356667, −10.34845302602962230917056787120, −8.297349492771777012058155360717, −6.95690046486759300501863547264, −6.35279619157414014039961646019, −3.84373025235866542455371398242, −2.00881966429327233362754463492, −0.23646546828919226880242572302,
1.92684545887187183042876449132, 4.26679875204560486708300887136, 5.55273225481550507523677798130, 6.68085404972383599435934066885, 9.251838957678566805551399268894, 9.840342877547128892123349277596, 11.26723447170717336209691061933, 12.05872453330523371814897724858, 14.11548130873210216183695440846, 15.05385381328855992225276995671
