| L(s) = 1 | + 0.792·3-s + i·5-s + (0.792 + 2.52i)7-s − 2.37·9-s − 0.792i·11-s + 5.37i·13-s + 0.792i·15-s + 3.37i·17-s + 3.46·19-s + (0.627 + 2i)21-s − 1.87i·23-s − 25-s − 4.25·27-s − 5.37·29-s + 8.51·31-s + ⋯ |
| L(s) = 1 | + 0.457·3-s + 0.447i·5-s + (0.299 + 0.954i)7-s − 0.790·9-s − 0.238i·11-s + 1.49i·13-s + 0.204i·15-s + 0.817i·17-s + 0.794·19-s + (0.136 + 0.436i)21-s − 0.391i·23-s − 0.200·25-s − 0.819·27-s − 0.997·29-s + 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19103 + 0.954627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19103 + 0.954627i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.792 - 2.52i)T \) |
| good | 3 | \( 1 - 0.792T + 3T^{2} \) |
| 11 | \( 1 + 0.792iT - 11T^{2} \) |
| 13 | \( 1 - 5.37iT - 13T^{2} \) |
| 17 | \( 1 - 3.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 8.51T + 31T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 - 2.74iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 0.744iT - 61T^{2} \) |
| 67 | \( 1 + 6.63iT - 67T^{2} \) |
| 71 | \( 1 + 6.63iT - 71T^{2} \) |
| 73 | \( 1 - 2.74iT - 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 17.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.11iT - 97T^{2} \) |
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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
−11.15871780330331162744779668315, −9.964021733565200714398167361599, −8.989554821247115217532852079475, −8.523931283403547355494201607478, −7.45913656893164379496827278184, −6.32283972230676409857494647301, −5.55315259451073591079431535186, −4.21298435153084184692151385635, −2.98305896971452687825074632131, −1.98455337630517258130088171025,
0.852196764176956167594421133826, 2.67998007498232105714427320259, 3.73694088137488497715633539865, 4.99016556458775371605127226468, 5.80487789808952520827389354912, 7.30936714052034587178860416233, 7.85047050348669740663338772051, 8.754239598119543585691296179977, 9.701457296850864837353277549831, 10.50033496105915012548710564329
