| L(s) = 1 | + 1.46·3-s + 0.462·5-s − 0.323·7-s − 0.860·9-s + 5.58·11-s + 6.58·13-s + 0.676·15-s − 6.64·17-s − 2.64·19-s − 0.473·21-s + 4.86·23-s − 4.78·25-s − 5.64·27-s − 4.86·29-s + 6.46·31-s + 8.16·33-s − 0.149·35-s − 37-s + 9.62·39-s − 0.815·41-s − 1.87·43-s − 0.398·45-s − 1.11·47-s − 6.89·49-s − 9.72·51-s − 12.6·53-s + 2.58·55-s + ⋯ |
| L(s) = 1 | + 0.844·3-s + 0.206·5-s − 0.122·7-s − 0.286·9-s + 1.68·11-s + 1.82·13-s + 0.174·15-s − 1.61·17-s − 0.607·19-s − 0.103·21-s + 1.01·23-s − 0.957·25-s − 1.08·27-s − 0.902·29-s + 1.16·31-s + 1.42·33-s − 0.0252·35-s − 0.164·37-s + 1.54·39-s − 0.127·41-s − 0.285·43-s − 0.0593·45-s − 0.163·47-s − 0.985·49-s − 1.36·51-s − 1.74·53-s + 0.348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.755193603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.755193603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 5 | \( 1 - 0.462T + 5T^{2} \) |
| 7 | \( 1 + 0.323T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 41 | \( 1 + 0.815T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 + 0.128T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 8.04T + 71T^{2} \) |
| 73 | \( 1 - 3.58T + 73T^{2} \) |
| 79 | \( 1 + 3.78T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 1.05T + 97T^{2} \) |
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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
−11.48486625577246254733228686920, −11.08368274153542791459108705379, −9.547408148510392708352804041462, −8.869996196977424984266331586994, −8.293683295342721729660434166900, −6.71769660084259130592038541006, −6.07682723621157549923626312270, −4.27208244246088495069323457097, −3.34243059582778935043951132583, −1.75209515526012064562747615162,
1.75209515526012064562747615162, 3.34243059582778935043951132583, 4.27208244246088495069323457097, 6.07682723621157549923626312270, 6.71769660084259130592038541006, 8.293683295342721729660434166900, 8.869996196977424984266331586994, 9.547408148510392708352804041462, 11.08368274153542791459108705379, 11.48486625577246254733228686920
