| L(s) = 1 | + (14.6 − 1.65i)2-s + (1.69 − 1.06i)3-s + (−36.4 + 8.32i)4-s + (784. + 625. i)5-s + (23.1 − 18.4i)6-s + (313. − 1.37e3i)7-s + (−4.09e3 + 1.43e3i)8-s + (−2.84e3 + 5.90e3i)9-s + (1.25e4 + 7.88e3i)10-s + (2.25e4 + 7.90e3i)11-s + (−52.9 + 52.9i)12-s + (7.33e3 + 1.52e4i)13-s + (2.33e3 − 2.07e4i)14-s + (1.99e3 + 224. i)15-s + (−4.91e4 + 2.36e4i)16-s + (5.00e4 + 5.00e4i)17-s + ⋯ |
| L(s) = 1 | + (0.918 − 0.103i)2-s + (0.0209 − 0.0131i)3-s + (−0.142 + 0.0325i)4-s + (1.25 + 1.00i)5-s + (0.0178 − 0.0142i)6-s + (0.130 − 0.572i)7-s + (−0.999 + 0.349i)8-s + (−0.433 + 0.900i)9-s + (1.25 + 0.788i)10-s + (1.54 + 0.539i)11-s + (−0.00255 + 0.00255i)12-s + (0.256 + 0.533i)13-s + (0.0607 − 0.538i)14-s + (0.0394 + 0.00444i)15-s + (−0.750 + 0.361i)16-s + (0.599 + 0.599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.72709 + 1.28770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72709 + 1.28770i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (8.10e4 + 7.02e5i)T \) |
| good | 2 | \( 1 + (-14.6 + 1.65i)T + (249. - 56.9i)T^{2} \) |
| 3 | \( 1 + (-1.69 + 1.06i)T + (2.84e3 - 5.91e3i)T^{2} \) |
| 5 | \( 1 + (-784. - 625. i)T + (8.69e4 + 3.80e5i)T^{2} \) |
| 7 | \( 1 + (-313. + 1.37e3i)T + (-5.19e6 - 2.50e6i)T^{2} \) |
| 11 | \( 1 + (-2.25e4 - 7.90e3i)T + (1.67e8 + 1.33e8i)T^{2} \) |
| 13 | \( 1 + (-7.33e3 - 1.52e4i)T + (-5.08e8 + 6.37e8i)T^{2} \) |
| 17 | \( 1 + (-5.00e4 - 5.00e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + (-6.28e4 + 9.99e4i)T + (-7.36e9 - 1.53e10i)T^{2} \) |
| 23 | \( 1 + (1.16e5 + 1.46e5i)T + (-1.74e10 + 7.63e10i)T^{2} \) |
| 31 | \( 1 + (1.40e6 - 1.58e5i)T + (8.31e11 - 1.89e11i)T^{2} \) |
| 37 | \( 1 + (1.81e6 - 6.36e5i)T + (2.74e12 - 2.18e12i)T^{2} \) |
| 41 | \( 1 + (-4.23e5 + 4.23e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-5.96e5 + 5.29e6i)T + (-1.13e13 - 2.60e12i)T^{2} \) |
| 47 | \( 1 + (-1.80e6 + 5.15e6i)T + (-1.86e13 - 1.48e13i)T^{2} \) |
| 53 | \( 1 + (-2.74e6 + 3.44e6i)T + (-1.38e13 - 6.06e13i)T^{2} \) |
| 59 | \( 1 - 2.07e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + (1.56e7 - 9.81e6i)T + (8.31e13 - 1.72e14i)T^{2} \) |
| 67 | \( 1 + (-4.64e5 + 9.65e5i)T + (-2.53e14 - 3.17e14i)T^{2} \) |
| 71 | \( 1 + (-6.07e6 - 1.26e7i)T + (-4.02e14 + 5.04e14i)T^{2} \) |
| 73 | \( 1 + (1.06e6 + 1.19e5i)T + (7.86e14 + 1.79e14i)T^{2} \) |
| 79 | \( 1 + (1.13e7 + 3.24e7i)T + (-1.18e15 + 9.45e14i)T^{2} \) |
| 83 | \( 1 + (-9.48e6 - 4.15e7i)T + (-2.02e15 + 9.77e14i)T^{2} \) |
| 89 | \( 1 + (-9.20e7 + 1.03e7i)T + (3.83e15 - 8.75e14i)T^{2} \) |
| 97 | \( 1 + (-9.91e7 - 6.22e7i)T + (3.40e15 + 7.06e15i)T^{2} \) |
| 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
−14.76669717587101375041968009899, −14.08078296647222550131997007426, −13.46825163858180552226587149441, −11.81634758421093558393295172686, −10.44090224371040255327347988615, −9.076161405698575849815426438444, −6.87281435513943919760589089605, −5.57132646397937311598404296144, −3.85603382454038340979972409511, −2.08932512300115106700353224659,
1.14563513800346254231064333348, 3.54135216981569917653940500879, 5.40748150775934198978408184958, 6.07330038677711148251369586607, 8.938013791234071605246548006659, 9.453150769982561488069492696929, 11.88736524525686198210745873601, 12.73020321374435382033178222289, 14.01253102617664081751761965827, 14.60767710010492479502003638357
