| L(s) = 1 | + (−0.684 + 1.18i)2-s + (1.5 + 2.59i)3-s + (3.06 + 5.30i)4-s + (−2.5 + 4.33i)5-s − 4.10·6-s + (17.2 + 6.62i)7-s − 19.3·8-s + (−4.5 + 7.79i)9-s + (−3.42 − 5.92i)10-s + (−4.53 − 7.85i)11-s + (−9.19 + 15.9i)12-s − 2.69·13-s + (−19.6 + 15.9i)14-s − 15.0·15-s + (−11.2 + 19.5i)16-s + (−7.75 − 13.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.418i)2-s + (0.288 + 0.499i)3-s + (0.383 + 0.663i)4-s + (−0.223 + 0.387i)5-s − 0.279·6-s + (0.933 + 0.357i)7-s − 0.854·8-s + (−0.166 + 0.288i)9-s + (−0.108 − 0.187i)10-s + (−0.124 − 0.215i)11-s + (−0.221 + 0.383i)12-s − 0.0575·13-s + (−0.375 + 0.304i)14-s − 0.258·15-s + (−0.176 + 0.305i)16-s + (−0.110 − 0.191i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.624044 + 1.42347i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.624044 + 1.42347i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (-17.2 - 6.62i)T \) |
| good | 2 | \( 1 + (0.684 - 1.18i)T + (-4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.53 + 7.85i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 2.69T + 2.19e3T^{2} \) |
| 17 | \( 1 + (7.75 + 13.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (23.2 - 40.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (23.7 - 41.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 8.21T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-114. - 197. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-167. + 289. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 35.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 510.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-272. + 472. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-2.16 - 3.75i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-395. - 685. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-107. + 186. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (201. + 348. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 328.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-261. - 453. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-65.3 + 113. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 507.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (282. - 490. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−13.95567601173483224386180347160, −12.44620639427552418973453718546, −11.51392735161773626430496653051, −10.58040048602841740563971815089, −9.044603658198961683127199945374, −8.166848257997470596741944460104, −7.23520918014834653979159136131, −5.70216541309324782862983543151, −4.02901928757753137241814039211, −2.53022254681198202175888219895,
0.946342798995600140560875231429, 2.37831550533182383578903319775, 4.52069505487979650340555968236, 6.04418333974008959155377986857, 7.42446425275668150782998200118, 8.517885519934776256597331582630, 9.719806148182179878806465769114, 10.91993899200166801075386632452, 11.68950908687045244184335289087, 12.78393225624308155529888684446
