| L(s) = 1 | + 1.56·3-s + 3.56·5-s − 3.12·7-s − 0.561·9-s + 11-s − 5.12·13-s + 5.56·15-s + 2·17-s + 4·19-s − 4.87·21-s − 2.43·23-s + 7.68·25-s − 5.56·27-s − 5.12·29-s + 5.56·31-s + 1.56·33-s − 11.1·35-s − 7.56·37-s − 8·39-s − 1.12·41-s + 7.12·43-s − 2·45-s − 8·47-s + 2.75·49-s + 3.12·51-s + 12.2·53-s + 3.56·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 1.59·5-s − 1.18·7-s − 0.187·9-s + 0.301·11-s − 1.42·13-s + 1.43·15-s + 0.485·17-s + 0.917·19-s − 1.06·21-s − 0.508·23-s + 1.53·25-s − 1.07·27-s − 0.951·29-s + 0.998·31-s + 0.271·33-s − 1.88·35-s − 1.24·37-s − 1.28·39-s − 0.175·41-s + 1.08·43-s − 0.298·45-s − 1.16·47-s + 0.393·49-s + 0.437·51-s + 1.68·53-s + 0.480·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596217583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596217583\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.12T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 5.12T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−12.94271734524480019568020824227, −11.95807808088777166357464121456, −10.18858133013568882613164308847, −9.646442871646915206051804286823, −9.020780896193437269789753892536, −7.52664524083583875814486797951, −6.32735885917481562884865699575, −5.31391946909881888985542805591, −3.28991764683894872626382791050, −2.23661788532368053880895832618,
2.23661788532368053880895832618, 3.28991764683894872626382791050, 5.31391946909881888985542805591, 6.32735885917481562884865699575, 7.52664524083583875814486797951, 9.020780896193437269789753892536, 9.646442871646915206051804286823, 10.18858133013568882613164308847, 11.95807808088777166357464121456, 12.94271734524480019568020824227
