L(s) = 1 | + 2i·2-s − 4·4-s + 8i·7-s − 8i·8-s + 18·11-s − 8i·13-s − 16·14-s + 16·16-s + 15i·17-s − 23·19-s + 36i·22-s − 63i·23-s + 16·26-s − 32i·28-s − 156·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.431i·7-s − 0.353i·8-s + 0.493·11-s − 0.170i·13-s − 0.305·14-s + 0.250·16-s + 0.214i·17-s − 0.277·19-s + 0.348i·22-s − 0.571i·23-s + 0.120·26-s − 0.215i·28-s − 0.998·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 8iT - 343T^{2} \) |
| 11 | \( 1 - 18T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 15iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 23T + 6.85e3T^{2} \) |
| 23 | \( 1 + 63iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 156T + 2.43e4T^{2} \) |
| 31 | \( 1 + 85T + 2.97e4T^{2} \) |
| 37 | \( 1 - 74iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 246T + 6.89e4T^{2} \) |
| 43 | \( 1 - 190iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 288iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 177iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 792T + 2.05e5T^{2} \) |
| 61 | \( 1 + 907T + 2.26e5T^{2} \) |
| 67 | \( 1 + 322iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 270T + 3.57e5T^{2} \) |
| 73 | \( 1 + 254iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 771iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 198T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3iT - 9.12e5T^{2} \) |
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
−9.051030846765812384461340653480, −8.052334333130376691684436604482, −7.41291583584188488755137938927, −6.36773384814830232011755188541, −5.85453264262044858608782795585, −4.80965449919218334224360813860, −3.96820204405594301717472522322, −2.81121604093329720658235333156, −1.46750653432545916045186829165, 0,
1.24379373231064953816497059673, 2.26425049342068580096804271970, 3.49382543185591561441561098237, 4.15541631334822718937275134324, 5.18478995185512754107150884997, 6.12915323667719069860385701574, 7.17971365768151308770570603195, 7.88042298201006568000471323522, 9.055455832060237230404706592805