| L(s) = 1 | + (−1.89 − 8.29i)2-s + (14.6 + 7.05i)3-s + (−36.3 + 17.5i)4-s + (−19.7 − 86.7i)5-s + (30.7 − 134. i)6-s + (23.9 + 11.5i)7-s + (44.5 + 55.8i)8-s + (13.1 + 16.5i)9-s + (−681. + 328. i)10-s + (−69.7 + 87.5i)11-s − 656.·12-s + (−33.6 + 42.1i)13-s + (50.3 − 220. i)14-s + (321. − 1.40e3i)15-s + (−427. + 535. i)16-s + 1.90e3·17-s + ⋯ |
| L(s) = 1 | + (−0.334 − 1.46i)2-s + (0.939 + 0.452i)3-s + (−1.13 + 0.547i)4-s + (−0.354 − 1.55i)5-s + (0.348 − 1.52i)6-s + (0.184 + 0.0889i)7-s + (0.245 + 0.308i)8-s + (0.0542 + 0.0679i)9-s + (−2.15 + 1.03i)10-s + (−0.173 + 0.218i)11-s − 1.31·12-s + (−0.0552 + 0.0692i)13-s + (0.0685 − 0.300i)14-s + (0.369 − 1.61i)15-s + (−0.417 + 0.523i)16-s + 1.60·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.364600 - 1.42597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.364600 - 1.42597i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-3.01e3 - 3.37e3i)T \) |
| good | 2 | \( 1 + (1.89 + 8.29i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-14.6 - 7.05i)T + (151. + 189. i)T^{2} \) |
| 5 | \( 1 + (19.7 + 86.7i)T + (-2.81e3 + 1.35e3i)T^{2} \) |
| 7 | \( 1 + (-23.9 - 11.5i)T + (1.04e4 + 1.31e4i)T^{2} \) |
| 11 | \( 1 + (69.7 - 87.5i)T + (-3.58e4 - 1.57e5i)T^{2} \) |
| 13 | \( 1 + (33.6 - 42.1i)T + (-8.26e4 - 3.61e5i)T^{2} \) |
| 17 | \( 1 - 1.90e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.11e3 + 535. i)T + (1.54e6 - 1.93e6i)T^{2} \) |
| 23 | \( 1 + (-755. + 3.30e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 31 | \( 1 + (-1.81e3 - 7.94e3i)T + (-2.57e7 + 1.24e7i)T^{2} \) |
| 37 | \( 1 + (-8.05e3 - 1.01e4i)T + (-1.54e7 + 6.76e7i)T^{2} \) |
| 41 | \( 1 + 1.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-4.23e3 + 1.85e4i)T + (-1.32e8 - 6.37e7i)T^{2} \) |
| 47 | \( 1 + (5.26e3 - 6.60e3i)T + (-5.10e7 - 2.23e8i)T^{2} \) |
| 53 | \( 1 + (-5.27e3 - 2.31e4i)T + (-3.76e8 + 1.81e8i)T^{2} \) |
| 59 | \( 1 + 2.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (3.31e4 + 1.59e4i)T + (5.26e8 + 6.60e8i)T^{2} \) |
| 67 | \( 1 + (1.10e4 + 1.38e4i)T + (-3.00e8 + 1.31e9i)T^{2} \) |
| 71 | \( 1 + (-1.92e3 + 2.40e3i)T + (-4.01e8 - 1.75e9i)T^{2} \) |
| 73 | \( 1 + (1.21e4 - 5.34e4i)T + (-1.86e9 - 8.99e8i)T^{2} \) |
| 79 | \( 1 + (-2.59e4 - 3.25e4i)T + (-6.84e8 + 2.99e9i)T^{2} \) |
| 83 | \( 1 + (4.20e4 - 2.02e4i)T + (2.45e9 - 3.07e9i)T^{2} \) |
| 89 | \( 1 + (1.17e4 + 5.14e4i)T + (-5.03e9 + 2.42e9i)T^{2} \) |
| 97 | \( 1 + (-4.74e4 + 2.28e4i)T + (5.35e9 - 6.71e9i)T^{2} \) |
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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
−15.59252717160308752451959924115, −14.06511338411532929603559769717, −12.60360020487914079732915575308, −11.94522444210462792518671425693, −10.17864793335061022780259910913, −9.100811672041284968737090322439, −8.291507400476459070366894685322, −4.70464474996581257817055261243, −3.13267944865620928343100198230, −1.05624886425019735164080358433,
3.03384108962515199343928709848, 5.94680876758117047181010908110, 7.59473681409529701683253577843, 7.78934308992643767759497723128, 9.676363317882975419665777568090, 11.43488207263557019791074605895, 13.68464580886464387068023523780, 14.48936768290330587799723047142, 15.11653077902391466731188141423, 16.40464916402718464303685563949
