| L(s) = 1 | + (1.96 + 8.60i)2-s + (176. + 163. i)3-s + (391. − 188. i)4-s + (58.9 − 8.88i)5-s + (−1.06e3 + 1.83e3i)6-s + (−1.46e3 − 2.54e3i)7-s + (5.20e3 + 6.52e3i)8-s + (2.84e3 + 3.79e4i)9-s + (192. + 489. i)10-s + (2.77e4 + 1.33e4i)11-s + (9.96e4 + 3.07e4i)12-s + (−3.50e4 + 8.92e4i)13-s + (1.89e4 − 1.76e4i)14-s + (1.18e4 + 8.07e3i)15-s + (9.26e4 − 1.16e5i)16-s + (1.97e5 + 2.97e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0867 + 0.380i)2-s + (1.25 + 1.16i)3-s + (0.763 − 0.367i)4-s + (0.0421 − 0.00635i)5-s + (−0.334 + 0.578i)6-s + (−0.230 − 0.400i)7-s + (0.449 + 0.563i)8-s + (0.144 + 1.92i)9-s + (0.00607 + 0.0154i)10-s + (0.571 + 0.275i)11-s + (1.38 + 0.428i)12-s + (−0.340 + 0.867i)13-s + (0.132 − 0.122i)14-s + (0.0603 + 0.0411i)15-s + (0.353 − 0.443i)16-s + (0.572 + 0.0863i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0481 - 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0481 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.61399 + 2.74292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.61399 + 2.74292i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.21e7 - 3.72e6i)T \) |
| good | 2 | \( 1 + (-1.96 - 8.60i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-176. - 163. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-58.9 + 8.88i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (1.46e3 + 2.54e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.77e4 - 1.33e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (3.50e4 - 8.92e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-1.97e5 - 2.97e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-6.73e3 + 8.98e4i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-2.56e5 + 1.74e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-7.59e5 + 7.04e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (-5.12e6 - 1.58e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (3.40e6 - 5.89e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (4.49e6 + 1.97e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (3.87e7 - 1.86e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (3.19e7 + 8.13e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-9.78e6 + 1.22e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-6.23e7 + 1.92e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-7.65e6 + 1.02e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (4.60e7 + 3.14e7i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (-1.56e8 + 3.98e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-1.67e8 - 2.89e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (5.49e7 + 5.10e7i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-3.47e7 - 3.22e7i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-3.94e8 - 1.90e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.49230253476306353751373724407, −13.72598871250134112200537450199, −11.70517638419663961067215967148, −10.29083583908872610966285179137, −9.515959153666917790202399220815, −8.100075999564824542881851883983, −6.73482159992411801681677239594, −4.83934316125261320071796054382, −3.45167723016793724552367112117, −1.94375018088634964816058530459,
1.21239804952656232743656374408, 2.49132915521850425565937332106, 3.42335822529652758528687966389, 6.27757501520955020873233833046, 7.49025329734639197595632959388, 8.398339302733047575830397417448, 9.902332093870514797514842304004, 11.72493257512305688471796239866, 12.55199860616314578444764573038, 13.47650363393276636663704437167
