| L(s) = 1 | + i·2-s − 4-s + 2.99i·5-s + (−0.713 − 2.54i)7-s − i·8-s − 2.99·10-s − 1.70·11-s − 1.66·13-s + (2.54 − 0.713i)14-s + 16-s + 4.46i·17-s + (−2.13 − 3.79i)19-s − 2.99i·20-s − 1.70i·22-s − 0.487·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.34i·5-s + (−0.269 − 0.962i)7-s − 0.353i·8-s − 0.947·10-s − 0.512·11-s − 0.461·13-s + (0.680 − 0.190i)14-s + 0.250·16-s + 1.08i·17-s + (−0.490 − 0.871i)19-s − 0.670i·20-s − 0.362i·22-s − 0.101·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6732400064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6732400064\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.713 + 2.54i)T \) |
| 19 | \( 1 + (2.13 + 3.79i)T \) |
| good | 5 | \( 1 - 2.99iT - 5T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 - 4.46iT - 17T^{2} \) |
| 23 | \( 1 + 0.487T + 23T^{2} \) |
| 29 | \( 1 + 1.42iT - 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 + 9.68iT - 37T^{2} \) |
| 41 | \( 1 + 3.63T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 + 2.80iT - 47T^{2} \) |
| 53 | \( 1 + 8.53iT - 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 + 3.96iT - 61T^{2} \) |
| 67 | \( 1 - 2.67iT - 67T^{2} \) |
| 71 | \( 1 + 8.53iT - 71T^{2} \) |
| 73 | \( 1 - 3.63iT - 73T^{2} \) |
| 79 | \( 1 + 14.7iT - 79T^{2} \) |
| 83 | \( 1 + 1.60iT - 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 - 4.57T + 97T^{2} \) |
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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
−8.686295215406848137512089585536, −7.88339575665514724785274769855, −7.18362291111494265396365034586, −6.70267376589185793459022890771, −6.02147717642654387149898583547, −4.94833702530028348819584769105, −4.00137279126606048630562372486, −3.24066193032821393083513884251, −2.14338285899493733993531505597, −0.24383783033139345147093537263,
1.15261176026760332503095607523, 2.28703035289035338282374478875, 3.13388849690591715987568039380, 4.36688960337694227892149941112, 5.05829120872393376399778511731, 5.59091068198489352053896866449, 6.66480747881903684337607000035, 7.900157569189124250991470182802, 8.473274019021892602351784933595, 9.078017864829940199518839407711
