| L(s) = 1 | + (−1.25 + 0.660i)2-s + i·3-s + (1.12 − 1.65i)4-s + i·5-s + (−0.660 − 1.25i)6-s − 3.91·7-s + (−0.319 + 2.81i)8-s − 9-s + (−0.660 − 1.25i)10-s − 3.56·11-s + (1.65 + 1.12i)12-s − 6.21·13-s + (4.89 − 2.58i)14-s − 15-s + (−1.45 − 3.72i)16-s − 6.87i·17-s + ⋯ |
| L(s) = 1 | + (−0.884 + 0.466i)2-s + 0.577i·3-s + (0.563 − 0.825i)4-s + 0.447i·5-s + (−0.269 − 0.510i)6-s − 1.47·7-s + (−0.113 + 0.993i)8-s − 0.333·9-s + (−0.208 − 0.395i)10-s − 1.07·11-s + (0.476 + 0.325i)12-s − 1.72·13-s + (1.30 − 0.690i)14-s − 0.258·15-s + (−0.363 − 0.931i)16-s − 1.66i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5184266654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5184266654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 - 0.660i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-3.05 - 3.69i)T \) |
| good | 7 | \( 1 + 3.91T + 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 6.21T + 13T^{2} \) |
| 17 | \( 1 + 6.87iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 5.49iT - 31T^{2} \) |
| 37 | \( 1 - 1.30iT - 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 - 2.84iT - 47T^{2} \) |
| 53 | \( 1 + 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 5.12iT - 59T^{2} \) |
| 61 | \( 1 + 5.14iT - 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 11.6iT - 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 0.545T + 79T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 + 7.41iT - 89T^{2} \) |
| 97 | \( 1 - 7.95iT - 97T^{2} \) |
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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
−9.552087137232045022704850172398, −9.197275942459687919826371209242, −7.76321599169544039086788263360, −7.27316016231906281335609653707, −6.56158425328396503715107280652, −5.40905737614593102863560064013, −4.89412029789049950645329104507, −2.92781759368785903357850619989, −2.78643626313041323476418223469, −0.40443395217339266020672605536,
0.78101166334603637305093533363, 2.41945962074370690298180969998, 2.95511358595067986342681110055, 4.27738468224838009211180338522, 5.62974836765263158933089017930, 6.50291278951434921252858999055, 7.38862054833783666852997497041, 7.907418860356323653001831454670, 8.873455641246020360837922908038, 9.640329492024117027418991823827
