| L(s) = 1 | + (4.85 + 0.425i)2-s + (−4.90 − 1.72i)3-s + (15.5 + 2.74i)4-s + (−3.76 + 10.5i)5-s + (−23.0 − 10.4i)6-s + (22.5 + 15.7i)7-s + (36.7 + 9.84i)8-s + (21.0 + 16.8i)9-s + (−22.7 + 49.5i)10-s + (8.26 − 22.7i)11-s + (−71.5 − 40.2i)12-s + (1.15 + 13.2i)13-s + (102. + 86.1i)14-s + (36.5 − 45.1i)15-s + (55.6 + 20.2i)16-s + (85.9 − 23.0i)17-s + ⋯ |
| L(s) = 1 | + (1.71 + 0.150i)2-s + (−0.943 − 0.331i)3-s + (1.94 + 0.342i)4-s + (−0.336 + 0.941i)5-s + (−1.57 − 0.711i)6-s + (1.21 + 0.851i)7-s + (1.62 + 0.435i)8-s + (0.779 + 0.625i)9-s + (−0.719 + 1.56i)10-s + (0.226 − 0.622i)11-s + (−1.72 − 0.968i)12-s + (0.0246 + 0.281i)13-s + (1.96 + 1.64i)14-s + (0.629 − 0.776i)15-s + (0.869 + 0.316i)16-s + (1.22 − 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.17040 + 1.27526i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.17040 + 1.27526i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.90 + 1.72i)T \) |
| 5 | \( 1 + (3.76 - 10.5i)T \) |
| good | 2 | \( 1 + (-4.85 - 0.425i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (-22.5 - 15.7i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-8.26 + 22.7i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-1.15 - 13.2i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-85.9 + 23.0i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (83.4 - 48.1i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (16.7 + 23.9i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (-60.0 + 50.3i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (16.3 - 92.9i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (56.5 + 211. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-126. + 151. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (237. + 509. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (331. - 473. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (-179. + 179. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-769. + 280. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (5.98 + 33.9i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-455. + 39.8i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-746. - 430. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (100. - 373. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (583. + 695. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-56.7 + 649. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-55.3 - 95.8i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (426. - 198. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.72151358030535393076544866478, −11.88017998620128617068752373228, −11.43070333076767290988295666185, −10.49417117006399299927019777782, −8.142195669222774706479221769697, −6.96241557808866149075248377457, −5.95798984908190148137811537174, −5.16348092181753790161068509044, −3.83928983718706765004073540525, −2.16561458118631410105404707123,
1.36333517398662589506847651147, 3.91174318355786864410201926913, 4.67535799099325124401634627060, 5.37182793793539876588305462537, 6.75158218052639766380749581584, 8.051054562739447048983519577466, 10.01725892007152506972075383260, 11.14457809180856629843430922185, 11.78922030589644247940155206436, 12.62774359183650054259494300738
