| L(s) = 1 | + (−0.987 + 0.717i)2-s + (−39.0 + 120. i)4-s + (339. + 246. i)5-s + (312. − 962. i)7-s + (−96.0 − 295. i)8-s − 512.·10-s + (1.00e3 − 4.29e3i)11-s + (1.04e4 − 7.61e3i)13-s + (381. + 1.17e3i)14-s + (−1.27e4 − 9.29e3i)16-s + (5.13e3 + 3.72e3i)17-s + (−6.62e3 − 2.03e4i)19-s + (−4.29e4 + 3.11e4i)20-s + (2.08e3 + 4.96e3i)22-s + 1.93e4·23-s + ⋯ |
| L(s) = 1 | + (−0.0873 + 0.0634i)2-s + (−0.305 + 0.939i)4-s + (1.21 + 0.882i)5-s + (0.344 − 1.06i)7-s + (−0.0663 − 0.204i)8-s − 0.162·10-s + (0.228 − 0.973i)11-s + (1.32 − 0.961i)13-s + (0.0371 + 0.114i)14-s + (−0.780 − 0.567i)16-s + (0.253 + 0.184i)17-s + (−0.221 − 0.682i)19-s + (−1.20 + 0.872i)20-s + (0.0418 + 0.0995i)22-s + 0.332·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0597i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.43566 + 0.0728874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43566 + 0.0728874i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-1.00e3 + 4.29e3i)T \) |
| good | 2 | \( 1 + (0.987 - 0.717i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-339. - 246. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-312. + 962. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-1.04e4 + 7.61e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-5.13e3 - 3.72e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (6.62e3 + 2.03e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 1.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (6.58e3 - 2.02e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-5.12e4 + 3.72e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-3.13e4 + 9.63e4i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.74e5 + 5.36e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 7.24e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.78e5 - 5.49e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (5.94e5 - 4.31e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (1.79e5 - 5.51e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.52e4 + 1.10e4i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 1.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-2.46e5 - 1.79e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (4.30e4 - 1.32e5i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (4.50e6 - 3.27e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-7.81e6 - 5.68e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.27e7 - 9.29e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.82558892427787607885769998596, −11.09787518291314250849983627048, −10.57326226927704694839317676021, −9.180128852839696262490941015809, −8.064447346966748244946346826266, −6.88683929413020281698437153052, −5.75547739914056744388404372188, −3.90585486511303872524728598195, −2.82642339510282663287530811645, −0.916609708007549204770986797493,
1.30735198850044153258625910322, 2.00073459459193248276308455039, 4.55007174314951496263167507381, 5.56602698732181692011356328873, 6.37757166416771871390973255560, 8.572726660652412872114379690477, 9.255083381142146849363027163141, 10.04082851137888189055928604930, 11.42013696813537992796611561560, 12.58665114916403582852093881015
