| L(s) = 1 | − 1.56·3-s + 3.40·5-s + 2.50·7-s − 0.561·9-s − 1.40·11-s + 6.22·13-s − 5.31·15-s − 3.16·17-s + 19-s − 3.91·21-s − 7.91·23-s + 6.60·25-s + 5.56·27-s + 6.22·29-s + 8.13·31-s + 2.19·33-s + 8.52·35-s − 6.13·37-s − 9.71·39-s + 1.68·41-s + 1.71·43-s − 1.91·45-s + 9.84·47-s − 0.729·49-s + 4.94·51-s + 2.78·53-s − 4.78·55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 1.52·5-s + 0.946·7-s − 0.187·9-s − 0.423·11-s + 1.72·13-s − 1.37·15-s − 0.767·17-s + 0.229·19-s − 0.853·21-s − 1.64·23-s + 1.32·25-s + 1.07·27-s + 1.15·29-s + 1.46·31-s + 0.382·33-s + 1.44·35-s − 1.00·37-s − 1.55·39-s + 0.263·41-s + 0.261·43-s − 0.285·45-s + 1.43·47-s − 0.104·49-s + 0.691·51-s + 0.382·53-s − 0.645·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551986931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551986931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 1.40T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 - 8.13T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 1.71T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 - 6.52T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 1.80T + 83T^{2} \) |
| 89 | \( 1 + 7.57T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−10.58121995719081733323303257963, −10.11478779715129631609204432598, −8.785286829678264472893947407320, −8.257367345852384471935595244817, −6.67379065150857517696529135388, −5.95782901618276610151341240399, −5.42583048296308275391962889041, −4.31576405370537100327769904158, −2.51245460756517635952807766016, −1.28544118175343581401372569750,
1.28544118175343581401372569750, 2.51245460756517635952807766016, 4.31576405370537100327769904158, 5.42583048296308275391962889041, 5.95782901618276610151341240399, 6.67379065150857517696529135388, 8.257367345852384471935595244817, 8.785286829678264472893947407320, 10.11478779715129631609204432598, 10.58121995719081733323303257963
