| L(s) = 1 | + (−1.41 − 2.44i)2-s + (−1.76 − 1.01i)3-s + (−3.99 + 6.92i)4-s + (0.615 − 0.355i)5-s + 5.75i·6-s + 22.6·8-s + (−38.4 − 66.5i)9-s + (−1.74 − 1.00i)10-s + (−75.8 + 131. i)11-s + (14.0 − 8.13i)12-s + 260. i·13-s − 1.44·15-s + (−32.0 − 55.4i)16-s + (333. + 192. i)17-s + (−108. + 188. i)18-s + (337. − 195. i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.195 − 0.112i)3-s + (−0.249 + 0.433i)4-s + (0.0246 − 0.0142i)5-s + 0.159i·6-s + 0.353·8-s + (−0.474 − 0.821i)9-s + (−0.0174 − 0.0100i)10-s + (−0.626 + 1.08i)11-s + (0.0978 − 0.0564i)12-s + 1.54i·13-s − 0.00642·15-s + (−0.125 − 0.216i)16-s + (1.15 + 0.667i)17-s + (−0.335 + 0.581i)18-s + (0.935 − 0.540i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.834692 + 0.381568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.834692 + 0.381568i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 2.44i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (1.76 + 1.01i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-0.615 + 0.355i)T + (312.5 - 541. i)T^{2} \) |
| 11 | \( 1 + (75.8 - 131. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 - 260. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (-333. - 192. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-337. + 195. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-88.8 - 153. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + 320.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-1.16e3 - 673. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-398. - 690. i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 - 815. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (3.71e3 - 2.14e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-1.58e3 + 2.74e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (4.07e3 + 2.35e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.19e3 - 1.26e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.04e3 + 3.54e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 2.25e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-4.02e3 - 2.32e3i)T + (1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-2.09e3 - 3.63e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 7.79e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-8.20e3 + 4.73e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 - 9.94e3iT - 8.85e7T^{2} \) |
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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
−13.10767797170317821830251028838, −12.02448246573413577707326641212, −11.44081378738775386771740143607, −9.956212908363169841899265673836, −9.273007240346786932709246294196, −7.84902367573347857552860108218, −6.60083174725603743276021762915, −4.92574434379567577573928109109, −3.31517397828956673186767997092, −1.52098869759977446587018162467,
0.51585276916174720844642983813, 3.02144915429939788057242030744, 5.20832967943992972809723294692, 5.89294647402730758235724124086, 7.75875422610234212705532841266, 8.238370057777245001434403514714, 9.892246833434625717530366912401, 10.66806759325651276440142016053, 11.88507216785691324375046955834, 13.36423324393651615284798523578
