| L(s) = 1 | + i·2-s − 4-s − 1.41i·5-s + (2.12 + 1.58i)7-s − i·8-s + 1.41·10-s + (−3.16 − i)11-s + 1.41·13-s + (−1.58 + 2.12i)14-s + 16-s + 4.47·17-s − 2.82·19-s + 1.41i·20-s + (1 − 3.16i)22-s + 3.16·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.632i·5-s + (0.801 + 0.597i)7-s − 0.353i·8-s + 0.447·10-s + (−0.953 − 0.301i)11-s + 0.392·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s + 1.08·17-s − 0.648·19-s + 0.316i·20-s + (0.213 − 0.674i)22-s + 0.659·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.711769888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.711769888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.12 - 1.58i)T \) |
| 11 | \( 1 + (3.16 + i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 + 6.32iT - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 3.16iT - 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 4.47iT - 97T^{2} \) |
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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
−9.409706808526907335744746938043, −8.533351750412881939240286983287, −8.230777806008799796302143662606, −7.35278537406988330106471913395, −6.27435924424503758045285986380, −5.30646926428312521475517036395, −5.02015004337449998262924188868, −3.79081207838627519517447552999, −2.46853941339923597480282697240, −0.968408047321580516110211522694,
1.02005019742427761440799917705, 2.32605803515588607782805164847, 3.25328635864286671133088062491, 4.30299848158619682078719683758, 5.11520333876029042582556692913, 6.11531501970781902499861502085, 7.31676719180158944267076551690, 7.83801923513746757848473119508, 8.713076655048789388026718667757, 9.766879915307259207151555116724
