L(s) = 1 | + 0.288i·2-s − i·3-s + 1.91·4-s + (−0.407 − 2.19i)5-s + 0.288·6-s + 3.66i·7-s + 1.13i·8-s − 9-s + (0.635 − 0.117i)10-s − 1.91i·12-s + 4.82i·13-s − 1.05·14-s + (−2.19 + 0.407i)15-s + 3.50·16-s − 3.84i·17-s − 0.288i·18-s + ⋯ |
L(s) = 1 | + 0.204i·2-s − 0.577i·3-s + 0.958·4-s + (−0.182 − 0.983i)5-s + 0.117·6-s + 1.38i·7-s + 0.399i·8-s − 0.333·9-s + (0.200 − 0.0371i)10-s − 0.553i·12-s + 1.33i·13-s − 0.283·14-s + (−0.567 + 0.105i)15-s + 0.876·16-s − 0.931i·17-s − 0.0680i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.205677669\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.205677669\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.407 + 2.19i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.288iT - 2T^{2} \) |
| 7 | \( 1 - 3.66iT - 7T^{2} \) |
| 13 | \( 1 - 4.82iT - 13T^{2} \) |
| 17 | \( 1 + 3.84iT - 17T^{2} \) |
| 19 | \( 1 - 6.96T + 19T^{2} \) |
| 23 | \( 1 + 1.20iT - 23T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 - 2.38iT - 37T^{2} \) |
| 41 | \( 1 - 4.30T + 41T^{2} \) |
| 43 | \( 1 - 0.772iT - 43T^{2} \) |
| 47 | \( 1 - 6.47iT - 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 6.10iT - 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 - 0.191iT - 73T^{2} \) |
| 79 | \( 1 - 3.15T + 79T^{2} \) |
| 83 | \( 1 - 1.53iT - 83T^{2} \) |
| 89 | \( 1 - 3.04T + 89T^{2} \) |
| 97 | \( 1 + 13.7iT - 97T^{2} \) |
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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
−9.206816484704027043318213048067, −8.453254466056979766231239032397, −7.65636572138969700462581214858, −6.97957942377244897574495206556, −6.02228640910562912909379695632, −5.47211524580735234403548430980, −4.55695757270829461605259114132, −3.04998016487840279317892210308, −2.21587230567589963558775127632, −1.22264650866365560691551685666,
0.929053592976226875646730022630, 2.45964808895943751299157815485, 3.51097777950906810759715355396, 3.75945337494639223144083083353, 5.26280477043297122621414573520, 6.10351035937111089567844940389, 7.00186964324766124896815345010, 7.58013639608597322283537794338, 8.162172311639927945603499500460, 9.687952290817158510245835935833