| L(s) = 1 | + i·3-s + 1.68i·5-s + 1.18·7-s − 9-s + 2.87i·11-s − i·13-s − 1.68·15-s − 6.70·17-s − 1.94i·19-s + 1.18i·21-s − 4.47·23-s + 2.14·25-s − i·27-s + 4.76i·29-s − 1.18·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.755i·5-s + 0.447·7-s − 0.333·9-s + 0.866i·11-s − 0.277i·13-s − 0.435·15-s − 1.62·17-s − 0.446i·19-s + 0.258i·21-s − 0.932·23-s + 0.429·25-s − 0.192i·27-s + 0.884i·29-s − 0.212·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6884524084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6884524084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 13 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.68iT - 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 11 | \( 1 - 2.87iT - 11T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + 1.94iT - 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 4.76iT - 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 2.70iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 2.19iT - 53T^{2} \) |
| 59 | \( 1 + 6.97iT - 59T^{2} \) |
| 61 | \( 1 - 11.5iT - 61T^{2} \) |
| 67 | \( 1 + 7.34iT - 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.87iT - 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−9.385234730774713739623137188513, −8.605257292903054580970772808840, −7.86814664616708291869051923406, −6.84673586884844158254485782575, −6.49548115955382167370665473149, −5.19079943729021131355975525999, −4.65657604022874657877072660640, −3.72746236465200223990534722059, −2.73250394867684534260541489640, −1.81728113979895670692453720126,
0.21547945878413641756044227389, 1.51176543417497264095849854732, 2.39258760990834569538479437182, 3.69323164317026980484767568850, 4.57307430928973548089611835546, 5.37691021148507308430224739419, 6.26612366661356420304185198588, 6.86737318622868848416025916529, 8.064706160163851052353042636519, 8.314593365574250322608833602737
