| L(s) = 1 | + (5.51 + 4.00i)2-s + (−2.78 + 8.55i)3-s + (4.46 + 13.7i)4-s + (−62.5 + 45.4i)5-s + (−49.6 + 36.0i)6-s + (59.3 + 182. i)7-s + (36.9 − 113. i)8-s + (−65.5 − 47.6i)9-s − 527.·10-s + (385. + 111. i)11-s − 129.·12-s + (−216. − 157. i)13-s + (−404. + 1.24e3i)14-s + (−215. − 662. i)15-s + (1.03e3 − 750. i)16-s + (1.18e3 − 864. i)17-s + ⋯ |
| L(s) = 1 | + (0.974 + 0.708i)2-s + (−0.178 + 0.549i)3-s + (0.139 + 0.429i)4-s + (−1.11 + 0.813i)5-s + (−0.562 + 0.408i)6-s + (0.458 + 1.40i)7-s + (0.204 − 0.628i)8-s + (−0.269 − 0.195i)9-s − 1.66·10-s + (0.960 + 0.277i)11-s − 0.260·12-s + (−0.355 − 0.258i)13-s + (−0.551 + 1.69i)14-s + (−0.246 − 0.760i)15-s + (1.00 − 0.733i)16-s + (0.998 − 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.860974 + 1.81633i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.860974 + 1.81633i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.78 - 8.55i)T \) |
| 11 | \( 1 + (-385. - 111. i)T \) |
| good | 2 | \( 1 + (-5.51 - 4.00i)T + (9.88 + 30.4i)T^{2} \) |
| 5 | \( 1 + (62.5 - 45.4i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-59.3 - 182. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (216. + 157. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.18e3 + 864. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (727. - 2.23e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 - 2.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (2.11e3 + 6.51e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-4.80e3 - 3.48e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-165. - 509. i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.60e3 - 1.11e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-143. + 441. i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (5.39e3 + 3.92e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.14e4 + 3.52e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.71e4 + 1.24e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.33e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.54e4 + 2.57e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (8.74e3 + 2.69e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.99e4 - 2.90e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-3.41e4 + 2.48e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 8.88e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + (9.30e4 + 6.76e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.56751360742167170315756221331, −14.96371981275016348163957906042, −14.36636472322729903375887403601, −12.29853987402347090434984204375, −11.57020747177017006154329530871, −9.807175227872369746410129834216, −7.949786978419292794730807811603, −6.37739140955897831615941041800, −4.98419618189519350082388573618, −3.47466568352350916334579490898,
1.05573402878071172047895799736, 3.74804262059930843436450034125, 4.80842835004593528787932213581, 7.20190538926814750690249378957, 8.485200543553834057807744635922, 10.89268024330373025709130199172, 11.76389477555670431541094706710, 12.71630503362554271850054838717, 13.73165220282344697340968446711, 14.83108641644962277279338895567
