| L(s) = 1 | + (8 + 13.8i)2-s + (−85.2 − 147. i)3-s + (−127. + 221. i)4-s + (−867. − 1.50e3i)5-s + (1.36e3 − 2.36e3i)6-s − 2.87e3·7-s − 4.09e3·8-s + (−4.68e3 + 8.11e3i)9-s + (1.38e4 − 2.40e4i)10-s + 2.31e4·11-s + 4.36e4·12-s + (−4.78e4 + 8.28e4i)13-s + (−2.29e4 − 3.98e4i)14-s + (−1.47e5 + 2.56e5i)15-s + (−3.27e4 − 5.67e4i)16-s + (8.43e4 + 1.46e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.607 − 1.05i)3-s + (−0.249 + 0.433i)4-s + (−0.620 − 1.07i)5-s + (0.429 − 0.743i)6-s − 0.452·7-s − 0.353·8-s + (−0.237 + 0.412i)9-s + (0.438 − 0.760i)10-s + 0.476·11-s + 0.607·12-s + (−0.464 + 0.804i)13-s + (−0.160 − 0.277i)14-s + (−0.754 + 1.30i)15-s + (−0.125 − 0.216i)16-s + (0.244 + 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.239094 + 0.370273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239094 + 0.370273i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 19 | \( 1 + (-3.37e5 - 4.56e5i)T \) |
| good | 3 | \( 1 + (85.2 + 147. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (867. + 1.50e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 7 | \( 1 + 2.87e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.31e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (4.78e4 - 8.28e4i)T + (-5.30e9 - 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-8.43e4 - 1.46e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 23 | \( 1 + (4.01e5 - 6.96e5i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + (4.96e5 - 8.59e5i)T + (-7.25e12 - 1.25e13i)T^{2} \) |
| 31 | \( 1 + 4.45e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.83e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-9.92e6 - 1.71e7i)T + (-1.63e14 + 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.61e7 - 2.79e7i)T + (-2.51e14 + 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-2.21e7 + 3.84e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-3.48e7 + 6.04e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.29e7 - 2.23e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.82e7 - 6.62e7i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.78e7 - 3.09e7i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (2.44e7 + 4.23e7i)T + (-2.29e16 + 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-7.43e7 - 1.28e8i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.18e8 + 3.78e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 4.70e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (3.94e8 - 6.83e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.11e8 - 1.92e8i)T + (-3.80e17 + 6.58e17i)T^{2} \) |
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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
−14.60570651518354294265130423098, −13.23171580048950401713994980983, −12.36089835955972771938003710230, −11.72325111061571075041732534034, −9.380498074090438950903001905841, −7.954696291402244605639472220971, −6.83802011800940637986661413369, −5.57431918842823639351824808934, −3.99483437159976621410061176564, −1.31795381633769844133222010767,
0.16749062894081562426239628646, 2.90950268552546036938702666720, 4.08267168320959325615563417375, 5.54477908749020999062571836573, 7.22448205773561992329050468356, 9.437947656252287276280730201023, 10.51813754807609639270002678101, 11.22563310739679744794391474187, 12.40048681663166168250661842036, 14.05689289226329330132722673483
