L(s) = 1 | − 2i·2-s − 2·4-s + i·5-s − 3.60i·7-s + 2·10-s + 3i·11-s − 3.60i·13-s − 7.21·14-s − 4·16-s + 3.60·17-s − 7.21i·19-s − 2i·20-s + 6·22-s − 3.60·23-s − 25-s − 7.21·26-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 4-s + 0.447i·5-s − 1.36i·7-s + 0.632·10-s + 0.904i·11-s − 0.999i·13-s − 1.92·14-s − 16-s + 0.874·17-s − 1.65i·19-s − 0.447i·20-s + 1.27·22-s − 0.751·23-s − 0.200·25-s − 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + 3.60iT \) |
good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 7 | \( 1 + 3.60iT - 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 7.21iT - 19T^{2} \) |
| 23 | \( 1 + 3.60T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 - 7.21iT - 31T^{2} \) |
| 37 | \( 1 + 3.60iT - 37T^{2} \) |
| 41 | \( 1 + 11iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 - 7.21iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 3iT - 89T^{2} \) |
| 97 | \( 1 + 3.60iT - 97T^{2} \) |
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
−10.30901884846963533949701005825, −9.970829568441752463891179860230, −8.819201692302068712432987102114, −7.35449638481312468048724572557, −7.03152230156419155182142912956, −5.33320711136710335634204686467, −4.12506382401174225492552797058, −3.38007843452385266801304455355, −2.17691516096619175552700088884, −0.69912277573216465176782182821,
2.03689542962607304281943648936, 3.75120582976975299775343947977, 5.12302110017609101935014833741, 5.87180431971600677317119580184, 6.33746068979094599465136030378, 7.81293144968649727380666148945, 8.227990178235678705588444610132, 9.111269148381400098569743879221, 9.828921537344254434339188799164, 11.44201407773603800801142731519