| L(s) = 1 | + 3-s + 5-s − 4.82·7-s + 9-s + 0.828·11-s + 2·13-s + 15-s + 5.65·17-s − 6.82·19-s − 4.82·21-s − 1.17·23-s + 25-s + 27-s − 6·29-s − 4·31-s + 0.828·33-s − 4.82·35-s − 7.65·37-s + 2·39-s − 3.65·41-s − 9.65·43-s + 45-s − 5.17·47-s + 16.3·49-s + 5.65·51-s + 9.31·53-s + 0.828·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.82·7-s + 0.333·9-s + 0.249·11-s + 0.554·13-s + 0.258·15-s + 1.37·17-s − 1.56·19-s − 1.05·21-s − 0.244·23-s + 0.200·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.144·33-s − 0.816·35-s − 1.25·37-s + 0.320·39-s − 0.571·41-s − 1.47·43-s + 0.149·45-s − 0.754·47-s + 2.33·49-s + 0.792·51-s + 1.27·53-s + 0.111·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.353928403589865262204718743251, −7.20645523713973161029720978273, −6.68967407576349151280366566906, −5.99077263816598197128166679878, −5.28678265527901992328333751890, −3.80506209521218478110385258156, −3.58721319899676426299426398163, −2.58981933385541535918782042168, −1.56778293428891148188984382172, 0,
1.56778293428891148188984382172, 2.58981933385541535918782042168, 3.58721319899676426299426398163, 3.80506209521218478110385258156, 5.28678265527901992328333751890, 5.99077263816598197128166679878, 6.68967407576349151280366566906, 7.20645523713973161029720978273, 8.353928403589865262204718743251
