| L(s) = 1 | − 2-s + 4-s − 2·5-s + 3.16i·7-s − 8-s + 3·9-s + 2·10-s + (1 − 3.16i)11-s − 6·13-s − 3.16i·14-s + 16-s + 6.32i·17-s − 3·18-s + (−3 + 3.16i)19-s − 2·20-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s + 1.19i·7-s − 0.353·8-s + 9-s + 0.632·10-s + (0.301 − 0.953i)11-s − 1.66·13-s − 0.845i·14-s + 0.250·16-s + 1.53i·17-s − 0.707·18-s + (−0.688 + 0.725i)19-s − 0.447·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.277065 + 0.469989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277065 + 0.469989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + (-1 + 3.16i)T \) |
| 19 | \( 1 + (3 - 3.16i)T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 3.16iT - 7T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 6.32iT - 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 9.48iT - 31T^{2} \) |
| 37 | \( 1 - 9.48iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 9.48iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 3.16iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 9.48iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.32iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 18.9iT - 97T^{2} \) |
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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
−11.61622458198183713081864071685, −10.37544247158519030771501800049, −9.812942474623324741811136710067, −8.404698546837973896382080257851, −8.254241647099399101113112800091, −6.95010365868690331727234701514, −6.02811862778622137859657311885, −4.64997904029601505456448773943, −3.34818067703718862826595783894, −1.84693101155797292146430206660,
0.43010836589856531599598196145, 2.32771420129432717051475941594, 4.10836519063235601385128301951, 4.70625924775662572057444076338, 6.74088728555677034406075891110, 7.44762265446535733044525620099, 7.71388039206146437273757543984, 9.401015651062948175712097107372, 9.869657420749556430021169861283, 10.73978347697589090123505274528
