| L(s) = 1 | + (−0.821 − 2.70i)2-s + (−1.42 − 9.90i)3-s + (−6.64 + 4.44i)4-s + (−9.08 − 2.66i)5-s + (−25.6 + 11.9i)6-s + (2.69 − 3.10i)7-s + (17.5 + 14.3i)8-s + (−70.2 + 20.6i)9-s + (0.248 + 26.7i)10-s + (11.8 + 18.3i)11-s + (53.5 + 59.5i)12-s + (36.8 − 31.9i)13-s + (−10.6 − 4.73i)14-s + (−13.4 + 93.8i)15-s + (24.4 − 59.1i)16-s + (−60.4 − 27.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.290 − 0.956i)2-s + (−0.274 − 1.90i)3-s + (−0.831 + 0.556i)4-s + (−0.812 − 0.238i)5-s + (−1.74 + 0.816i)6-s + (0.145 − 0.167i)7-s + (0.773 + 0.633i)8-s + (−2.60 + 0.763i)9-s + (0.00784 + 0.847i)10-s + (0.323 + 0.503i)11-s + (1.28 + 1.43i)12-s + (0.785 − 0.680i)13-s + (−0.202 − 0.0903i)14-s + (−0.232 + 1.61i)15-s + (0.381 − 0.924i)16-s + (−0.863 − 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.209593 + 0.0672695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209593 + 0.0672695i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.821 + 2.70i)T \) |
| 23 | \( 1 + (-51.2 + 97.6i)T \) |
| good | 3 | \( 1 + (1.42 + 9.90i)T + (-25.9 + 7.60i)T^{2} \) |
| 5 | \( 1 + (9.08 + 2.66i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-2.69 + 3.10i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-11.8 - 18.3i)T + (-552. + 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-36.8 + 31.9i)T + (312. - 2.17e3i)T^{2} \) |
| 17 | \( 1 + (60.4 + 27.6i)T + (3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (74.6 - 34.0i)T + (4.49e3 - 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-71.3 - 32.5i)T + (1.59e4 + 1.84e4i)T^{2} \) |
| 31 | \( 1 + (163. + 23.4i)T + (2.85e4 + 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-327. + 96.2i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (66.6 + 19.5i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-87.0 + 12.5i)T + (7.62e4 - 2.23e4i)T^{2} \) |
| 47 | \( 1 - 498. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (388. - 448. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-157. - 182. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-67.5 + 469. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (41.1 - 63.9i)T + (-1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (320. - 498. i)T + (-1.48e5 - 3.25e5i)T^{2} \) |
| 73 | \( 1 + (324. + 710. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (337. + 389. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (209. + 712. i)T + (-4.81e5 + 3.09e5i)T^{2} \) |
| 89 | \( 1 + (1.43e3 - 206. i)T + (6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (267. - 909. i)T + (-7.67e5 - 4.93e5i)T^{2} \) |
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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
−11.35783468600932271952752032450, −10.84663999028866617508699040746, −8.954356882437643051035839927411, −8.121361376114142714566914381556, −7.40026059555828205897847562131, −6.12241287299617265768749640589, −4.39288619871526320257308352560, −2.66548475951588367951921607811, −1.31122064487959892861659966612, −0.12402285465815187175660479132,
3.69787259073075460742301521201, 4.37340489511442967071473601316, 5.57353425910731465003682424643, 6.65403861058256529600815209978, 8.380018251538444463661093826451, 8.920347709248290194824353035993, 9.905440286220725848496524556915, 11.08161153281024832256117354640, 11.41260889176558432416207498493, 13.38127805932135639177547014539
