| L(s) = 1 | + (−1.61 − 1.17i)2-s + (−2.41 + 7.42i)3-s + (1.23 + 3.80i)4-s + (−12.0 + 8.76i)5-s + (12.6 − 9.18i)6-s + (−6.72 − 20.7i)7-s + (2.47 − 7.60i)8-s + (−27.5 − 19.9i)9-s + 29.8·10-s − 31.2·12-s + (−35.6 − 25.8i)13-s + (−13.4 + 41.4i)14-s + (−36.0 − 110. i)15-s + (−12.9 + 9.40i)16-s + (−20.1 + 14.6i)17-s + (21.0 + 64.7i)18-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.464 + 1.42i)3-s + (0.154 + 0.475i)4-s + (−1.07 + 0.784i)5-s + (0.860 − 0.624i)6-s + (−0.363 − 1.11i)7-s + (0.109 − 0.336i)8-s + (−1.01 − 0.740i)9-s + 0.943·10-s − 0.751·12-s + (−0.759 − 0.552i)13-s + (−0.256 + 0.790i)14-s + (−0.619 − 1.90i)15-s + (−0.202 + 0.146i)16-s + (−0.287 + 0.208i)17-s + (0.275 + 0.847i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.433701 - 0.119283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.433701 - 0.119283i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 + 1.17i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (2.41 - 7.42i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (12.0 - 8.76i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (6.72 + 20.7i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (35.6 + 25.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (20.1 - 14.6i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (6.78 - 20.8i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-46.1 - 142. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (60.7 + 44.1i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-68.7 - 211. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-78.2 + 240. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 130.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-154. + 474. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (10.4 + 7.60i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 33.8i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-435. + 316. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (63.4 - 46.1i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (353. + 1.08e3i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-625. - 454. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-434. + 315. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-145. - 105. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.12480241583974103395425930112, −10.60520980034222765872155634621, −10.05561057549076001767754349978, −8.938567350394667139103667936869, −7.60812145973518042734367682436, −6.82755257564044333244858519060, −4.98982223677615067906124750444, −3.88614376525483079089854479783, −3.17299929860716371719346218127, −0.32492445900651244857327913546,
0.880271554769411245808314451391, 2.46318663201911220594300645778, 4.72450974843648274960854785919, 5.90320354005555026931708878115, 6.94394976806200930349435560504, 7.68627669223274109018881295886, 8.654535999926163221068908258074, 9.399350039458256768748261608881, 11.19344167935986797390789346896, 11.87363125656854681047989764036
