| L(s) = 1 | − 1.15·5-s − 0.123·7-s − 0.794i·11-s − 0.814i·13-s − 4.10·17-s + 2.35·19-s − 4.07·23-s − 3.67·25-s − 4.97i·29-s + 8.23·31-s + 0.142·35-s − 4.60i·37-s − 1.20·41-s + 10.0i·43-s + 8.74i·47-s + ⋯ |
| L(s) = 1 | − 0.514·5-s − 0.0467·7-s − 0.239i·11-s − 0.225i·13-s − 0.996·17-s + 0.541·19-s − 0.848·23-s − 0.735·25-s − 0.923i·29-s + 1.47·31-s + 0.0240·35-s − 0.757i·37-s − 0.187·41-s + 1.53i·43-s + 1.27i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9637430835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9637430835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + (-9.57 - 8.68i)T \) |
| good | 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 + 0.123T + 7T^{2} \) |
| 11 | \( 1 + 0.794iT - 11T^{2} \) |
| 13 | \( 1 + 0.814iT - 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 + 4.97iT - 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 + 4.60iT - 37T^{2} \) |
| 41 | \( 1 + 1.20T + 41T^{2} \) |
| 43 | \( 1 - 10.0iT - 43T^{2} \) |
| 47 | \( 1 - 8.74iT - 47T^{2} \) |
| 53 | \( 1 - 6.14T + 53T^{2} \) |
| 59 | \( 1 - 5.67T + 59T^{2} \) |
| 61 | \( 1 - 5.76T + 61T^{2} \) |
| 67 | \( 1 + 1.63iT - 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + 7.10iT - 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 2.78iT - 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
| 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
−8.014801544569170765974424176496, −7.80821489894814853286306945319, −6.74872756265374890181036171039, −6.19459880162808468254111460627, −5.41413006710395863342711535545, −4.44743057712212409946360651187, −3.96898029441366306435281549016, −2.97092447277947749720368313058, −2.17106236392851443344054957446, −0.889148412764841695988016544585,
0.29876026325720128765903857348, 1.66150618710118146224431409535, 2.55586046188559702502043545127, 3.57437202201910995951031850859, 4.20206417633647888856263531502, 4.98506187144016694931285172578, 5.74897432278509200457567503315, 6.73307792995880527712683685318, 7.04386369906120662443112200682, 8.067853204773911776470971964628
