| L(s) = 1 | + (2.54 + 4.41i)2-s + (−3.72 + 3.62i)3-s + (−8.97 + 15.5i)4-s + 6.84·5-s + (−25.4 − 7.19i)6-s + (16.1 − 9.05i)7-s − 50.6·8-s + (0.730 − 26.9i)9-s + (17.4 + 30.1i)10-s − 41.0·11-s + (−22.9 − 90.4i)12-s + (31.8 + 55.2i)13-s + (81.0 + 48.2i)14-s + (−25.4 + 24.7i)15-s + (−57.3 − 99.2i)16-s + (38.0 + 65.8i)17-s + ⋯ |
| L(s) = 1 | + (0.900 + 1.55i)2-s + (−0.716 + 0.697i)3-s + (−1.12 + 1.94i)4-s + 0.611·5-s + (−1.73 − 0.489i)6-s + (0.872 − 0.488i)7-s − 2.24·8-s + (0.0270 − 0.999i)9-s + (0.550 + 0.954i)10-s − 1.12·11-s + (−0.551 − 2.17i)12-s + (0.680 + 1.17i)13-s + (1.54 + 0.920i)14-s + (−0.438 + 0.426i)15-s + (−0.895 − 1.55i)16-s + (0.542 + 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.140874 + 1.85893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.140874 + 1.85893i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.72 - 3.62i)T \) |
| 7 | \( 1 + (-16.1 + 9.05i)T \) |
| good | 2 | \( 1 + (-2.54 - 4.41i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 6.84T + 125T^{2} \) |
| 11 | \( 1 + 41.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + (-31.8 - 55.2i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-38.0 - 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.7 + 81.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 42.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-21.9 + 37.9i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-100. + 173. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (52.1 - 90.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (108. + 188. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-236. + 409. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-8.94 - 15.4i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-211. - 365. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (145. - 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-77.0 - 133. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-419. + 726. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 940.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-65.8 - 113. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (310. + 537. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-102. + 177. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-432. + 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-331. + 573. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.17949995154564677268355854657, −14.01431914303913093376656839062, −13.31741208234280241512831667402, −11.81179876325834790546857400749, −10.47457461735574589821643604688, −8.865604206543694139749698907355, −7.46027027097950502260709225980, −6.12800054771156496150651117702, −5.16974802926033222742607145625, −4.08330283441031813080182451300,
1.26222458764724315004823595925, 2.76359684448090948390787996248, 5.12316668284925817565092882199, 5.67756829096675718791177897072, 8.006565338725801077798134940430, 9.994319271059943895464505486201, 10.85748848926211330826187484549, 11.79418196047067777175573041305, 12.71093416269175815068626067060, 13.50998507721103507911280749603
