| L(s) = 1 | + 3i·3-s + 8·4-s + 2i·7-s − 9·9-s − 11·11-s + 24i·12-s + 22i·13-s + 64·16-s + 72i·17-s − 122·19-s − 6·21-s − 72i·23-s − 27i·27-s + 16i·28-s − 96·29-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 4-s + 0.107i·7-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s + 0.469i·13-s + 16-s + 1.02i·17-s − 1.47·19-s − 0.0623·21-s − 0.652i·23-s − 0.192i·27-s + 0.107i·28-s − 0.614·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.385401357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385401357\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 7 | \( 1 - 2iT - 343T^{2} \) |
| 13 | \( 1 - 22iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 72iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 122T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 96T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 96T + 6.89e4T^{2} \) |
| 43 | \( 1 - 382iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 360iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 318iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 660T + 2.05e5T^{2} \) |
| 61 | \( 1 + 430T + 2.26e5T^{2} \) |
| 67 | \( 1 - 380iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 218iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 706T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 686iT - 9.12e5T^{2} \) |
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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
−10.50210413317770532520602311530, −9.413171273882073663846105758330, −8.465639860056790505507724692842, −7.71038867048568857713098436136, −6.51814503178192843582106513828, −6.04219673770863734576177467077, −4.79129643749256951234640483491, −3.79220746392183930403233684744, −2.65849254525560447036006237603, −1.64281685125503533158083451526,
0.31687964756715890919229843694, 1.77881667053972672895781121368, 2.63230275770641676940415652563, 3.76970891299790811160157394364, 5.27896774890494054766908552370, 6.03973840526434426074794779326, 7.06051459126339102881501672752, 7.50696263722078808079534927356, 8.475754078358756440594072476480, 9.469336496769349918367958688757
