| L(s) = 1 | − 5.51i·2-s + (−1.33 + 0.771i)3-s − 14.3·4-s + (−29.9 + 17.2i)5-s + (4.24 + 7.36i)6-s + (−50.7 − 29.2i)7-s − 8.99i·8-s + (−39.3 + 68.0i)9-s + (95.3 + 165. i)10-s + 47.6·11-s + (19.1 − 11.0i)12-s + (88.6 − 153. i)13-s + (−161. + 279. i)14-s + (26.6 − 46.2i)15-s − 279.·16-s + (72.8 − 126. i)17-s + ⋯ |
| L(s) = 1 | − 1.37i·2-s + (−0.148 + 0.0856i)3-s − 0.897·4-s + (−1.19 + 0.691i)5-s + (0.118 + 0.204i)6-s + (−1.03 − 0.597i)7-s − 0.140i·8-s + (−0.485 + 0.840i)9-s + (0.953 + 1.65i)10-s + 0.393·11-s + (0.133 − 0.0769i)12-s + (0.524 − 0.909i)13-s + (−0.823 + 1.42i)14-s + (0.118 − 0.205i)15-s − 1.09·16-s + (0.251 − 0.436i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.137395 + 0.283364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137395 + 0.283364i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.54e3 - 1.01e3i)T \) |
| good | 2 | \( 1 + 5.51iT - 16T^{2} \) |
| 3 | \( 1 + (1.33 - 0.771i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (29.9 - 17.2i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (50.7 + 29.2i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 - 47.6T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-88.6 + 153. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (-72.8 + 126. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (339. - 196. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (518. + 898. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-972. - 561. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (260. + 451. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.40e3 - 811. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 1.03e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 2.18e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (526. + 912. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 - 3.84e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + (2.56e3 + 1.48e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-295. - 512. i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-4.49 - 2.59i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-2.19e3 - 1.26e3i)T + (1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-863. + 1.49e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (4.34e3 + 7.53e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.31e4 - 7.58e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 7.78e3T + 8.85e7T^{2} \) |
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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
−14.19932409817223459124111143709, −12.88046838983518784804540685127, −11.90660187882205387670982795515, −10.74632070473583835041435399172, −10.24723362939017750412241356528, −8.271203736734980408182192361608, −6.65579476261381674087800192892, −4.06069700834023434608451238025, −2.93245388855731040572118942490, −0.20011622255795076656622372039,
3.95441565358049166438132382815, 5.84768240684040288948724720647, 6.87007776527875225013762279074, 8.393586592591195987353931739960, 9.180622498825354617891036873106, 11.57556079864812746642444765934, 12.38897648162019648986202905500, 13.94362742259918552279676374221, 15.34779083730583068622449350413, 15.78081914602552713559499171875
