L(s) = 1 | − 2.82i·5-s + i·7-s + 1.41·11-s − 2·13-s + 7.07·23-s − 3.00·25-s − 9.89i·29-s − 4i·31-s + 2.82·35-s + 2.82i·41-s − 4i·43-s + 2.82·47-s − 49-s − 4.24i·53-s − 4.00i·55-s + ⋯ |
L(s) = 1 | − 1.26i·5-s + 0.377i·7-s + 0.426·11-s − 0.554·13-s + 1.47·23-s − 0.600·25-s − 1.83i·29-s − 0.718i·31-s + 0.478·35-s + 0.441i·41-s − 0.609i·43-s + 0.412·47-s − 0.142·49-s − 0.582i·53-s − 0.539i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495509261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495509261\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 9.89iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.175104463484998582447294120007, −8.203704512559838155648909425584, −7.53996416472029429360829836427, −6.48206157586021002695001319933, −5.64739558181282096177381646039, −4.83705498833167358416491705276, −4.22470355973473536538549117117, −2.95393654252162424218046187240, −1.78781076218911119458511871537, −0.56557605447525627040454183607,
1.35896686894928774431925936307, 2.77783392857995085703289123041, 3.32725408469917901749656852368, 4.46101054022409817401880442905, 5.37244848859719355475287577929, 6.45395562642449660141746778741, 7.06454845603752847662683550309, 7.48679082250974263271284539310, 8.704012257788808466940264483272, 9.334171452299861624061307629047