| L(s) = 1 | + 2.76·3-s − 1.62·5-s + 4.62·9-s + 4.49·11-s + 0.103·13-s − 4.49·15-s + 2·17-s − 7.25·19-s − 5.52·23-s − 2.35·25-s + 4.49·27-s + 29-s − 6.76·31-s + 12.4·33-s + 5.25·37-s + 0.284·39-s + 5.79·41-s − 10.0·43-s − 7.52·45-s + 11.5·47-s − 7·49-s + 5.52·51-s − 7.14·53-s − 7.30·55-s − 20.0·57-s − 1.52·59-s + 9.04·61-s + ⋯ |
| L(s) = 1 | + 1.59·3-s − 0.727·5-s + 1.54·9-s + 1.35·11-s + 0.0285·13-s − 1.15·15-s + 0.485·17-s − 1.66·19-s − 1.15·23-s − 0.471·25-s + 0.864·27-s + 0.185·29-s − 1.21·31-s + 2.15·33-s + 0.863·37-s + 0.0455·39-s + 0.904·41-s − 1.52·43-s − 1.12·45-s + 1.68·47-s − 49-s + 0.773·51-s − 0.982·53-s − 0.984·55-s − 2.65·57-s − 0.198·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807911200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807911200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 13 | \( 1 - 0.103T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 31 | \( 1 + 6.76T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 7.14T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 9.04T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
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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
−12.31761976635299702658453731049, −11.30624699566633253097347632845, −9.983605244143744973122604029609, −9.077208846541760005490152300460, −8.306243531812931932099951485565, −7.52315823895591231233999998668, −6.30242378575939293747318321251, −4.21650643530446016899124548191, −3.59309401555833095767376005753, −2.01856914864810775413025137690,
2.01856914864810775413025137690, 3.59309401555833095767376005753, 4.21650643530446016899124548191, 6.30242378575939293747318321251, 7.52315823895591231233999998668, 8.306243531812931932099951485565, 9.077208846541760005490152300460, 9.983605244143744973122604029609, 11.30624699566633253097347632845, 12.31761976635299702658453731049
