L(s) = 1 | + i·3-s + 2·4-s + 2·9-s − 4·11-s + 2i·12-s + 6i·13-s + 4·16-s + 5i·17-s + 19-s − 8i·23-s + 5i·27-s + 10·29-s − 31-s − 4i·33-s + 4·36-s + i·37-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 4-s + 0.666·9-s − 1.20·11-s + 0.577i·12-s + 1.66i·13-s + 16-s + 1.21i·17-s + 0.229·19-s − 1.66i·23-s + 0.962i·27-s + 1.85·29-s − 0.179·31-s − 0.696i·33-s + 0.666·36-s + 0.164i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64554 + 1.01700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64554 + 1.01700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 2T^{2} \) |
| 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 5iT - 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 9iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−10.48278345533446524992325430119, −9.904256373478041683973361706975, −8.722197186877822478329589311283, −7.88632026386161103566314793055, −6.83214982711235958026752157333, −6.28466943321559373227459233947, −4.94007675554633247580204355813, −4.12339276751089641210516377440, −2.81589308501884073096171315419, −1.70655469139794557059481133399,
1.03228321438965306959745188242, 2.48207714564487906477891693061, 3.25378305761747039797499486378, 5.00300429306524236642947842196, 5.76554771935643504569960634164, 6.84439527914552988698813881367, 7.70915822697217151511001502685, 7.890394711819109950625215206864, 9.466003611325508049518057608641, 10.41044303706651300394071596226