| L(s) = 1 | + 2i·2-s + 4i·3-s − 4·4-s − 8·6-s − 20i·7-s − 8i·8-s + 11·9-s − 44·11-s − 16i·12-s − 42i·13-s + 40·14-s + 16·16-s − 86i·17-s + 22i·18-s − 19·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.769i·3-s − 0.5·4-s − 0.544·6-s − 1.07i·7-s − 0.353i·8-s + 0.407·9-s − 1.20·11-s − 0.384i·12-s − 0.896i·13-s + 0.763·14-s + 0.250·16-s − 1.22i·17-s + 0.288i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.145520711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.145520711\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 4iT - 27T^{2} \) |
| 7 | \( 1 + 20iT - 343T^{2} \) |
| 11 | \( 1 + 44T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 86iT - 4.91e3T^{2} \) |
| 23 | \( 1 - 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 162T + 2.43e4T^{2} \) |
| 31 | \( 1 + 312T + 2.97e4T^{2} \) |
| 37 | \( 1 - 226iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 34T + 6.89e4T^{2} \) |
| 43 | \( 1 - 432iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 580iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 506iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364T + 2.05e5T^{2} \) |
| 61 | \( 1 - 518T + 2.26e5T^{2} \) |
| 67 | \( 1 - 924iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 320T + 3.57e5T^{2} \) |
| 73 | \( 1 - 542iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3iT - 9.12e5T^{2} \) |
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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
−9.882721140005250697826044375749, −9.401091254797735589634204044887, −8.062665340537514203129447348305, −7.55368864538176865495321368123, −6.78130466521080935894303728682, −5.42394012778622198073007603905, −4.91598918117742613558666823765, −3.92651949796314032549333748919, −2.96897344424761143363456060342, −1.01310994925174928615743872480,
0.33415302523609919438880253279, 1.95274250460059837051992414129, 2.29956341570017540144513040906, 3.75363955329574058965639566199, 4.84685854497597558147563005285, 5.82886638171804293756282590521, 6.72177131456271208666227611312, 7.75009936945931162017649807694, 8.581470023089136208719360511137, 9.183769785699725115909318318797
