L(s) = 1 | + 2i·3-s − 1.73i·5-s + 1.73·7-s − 9-s + 3i·11-s + 3.46·15-s − 3·17-s + i·19-s + 3.46i·21-s + 3.46·23-s + 2.00·25-s + 4i·27-s − 3.46i·29-s + 6.92·31-s − 6·33-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 0.774i·5-s + 0.654·7-s − 0.333·9-s + 0.904i·11-s + 0.894·15-s − 0.727·17-s + 0.229i·19-s + 0.755i·21-s + 0.722·23-s + 0.400·25-s + 0.769i·27-s − 0.643i·29-s + 1.24·31-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773590533\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773590533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.904260175451392310280789054761, −9.134651443908583858582983202941, −8.510258927790846306545128476937, −7.58103408563278711420364678001, −6.54061604236891968259855100544, −5.30996495404645124766230089083, −4.57629126388276573532465852878, −4.26152648992979132034159970546, −2.78433399964219420287668859552, −1.34366561065017568858271144626,
0.869532506126091502286124386544, 2.14464037007539385288493675998, 3.05900881158751192129317369803, 4.36641153913946972863328227426, 5.52567955327388282498926208982, 6.50430011219516939125375699798, 7.00394066361180618803073787617, 7.83560832820863925912158877143, 8.540428649786032302996854370281, 9.432763625990370157256677661334