| L(s) = 1 | − 2.61·3-s + 3.41·5-s − 1.53·7-s + 3.82·9-s + 4.77·11-s + 0.585·13-s − 8.92·15-s − 2.82·17-s + 0.448·19-s + 4·21-s − 5.86·23-s + 6.65·25-s − 2.16·27-s + 4.58·29-s + 7.39·31-s − 12.4·33-s − 5.22·35-s − 5.07·37-s − 1.53·39-s − 4·41-s + 2.61·43-s + 13.0·45-s + 7.39·47-s − 4.65·49-s + 7.39·51-s + 7.41·53-s + 16.3·55-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 1.52·5-s − 0.578·7-s + 1.27·9-s + 1.44·11-s + 0.162·13-s − 2.30·15-s − 0.685·17-s + 0.102·19-s + 0.872·21-s − 1.22·23-s + 1.33·25-s − 0.416·27-s + 0.851·29-s + 1.32·31-s − 2.17·33-s − 0.883·35-s − 0.833·37-s − 0.245·39-s − 0.624·41-s + 0.398·43-s + 1.94·45-s + 1.07·47-s − 0.665·49-s + 1.03·51-s + 1.01·53-s + 2.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.261448799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261448799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 0.448T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 4.58T + 29T^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 - 7.41T + 53T^{2} \) |
| 59 | \( 1 + 2.61T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 + 3.50T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−10.01151418964978744596009809977, −9.440896840295615116831456353651, −8.480430152060520178172646238244, −6.68372788424544699424761403689, −6.56165961618439836198526720966, −5.82626464540961838981061684282, −4.99352124326692809550077641306, −3.89480476744320597052986510397, −2.22630838807908370909208512995, −0.980969940577510576536911232947,
0.980969940577510576536911232947, 2.22630838807908370909208512995, 3.89480476744320597052986510397, 4.99352124326692809550077641306, 5.82626464540961838981061684282, 6.56165961618439836198526720966, 6.68372788424544699424761403689, 8.480430152060520178172646238244, 9.440896840295615116831456353651, 10.01151418964978744596009809977
