| L(s) = 1 | + 1.94·3-s − 3.94·5-s − 3.65·7-s + 0.774·9-s + 1.35·11-s + 6.24·13-s − 7.65·15-s + 2·17-s + 1.94·19-s − 7.10·21-s + 6.52·23-s + 10.5·25-s − 4.32·27-s − 2.77·29-s + 0.550·31-s + 2.64·33-s + 14.4·35-s − 7.35·37-s + 12.1·39-s − 5.65·41-s − 11.9·43-s − 3.05·45-s − 5.97·47-s + 6.39·49-s + 3.88·51-s − 3.94·53-s − 5.35·55-s + ⋯ |
| L(s) = 1 | + 1.12·3-s − 1.76·5-s − 1.38·7-s + 0.258·9-s + 0.409·11-s + 1.73·13-s − 1.97·15-s + 0.485·17-s + 0.445·19-s − 1.55·21-s + 1.36·23-s + 2.10·25-s − 0.832·27-s − 0.515·29-s + 0.0987·31-s + 0.459·33-s + 2.43·35-s − 1.20·37-s + 1.94·39-s − 0.883·41-s − 1.82·43-s − 0.455·45-s − 0.871·47-s + 0.913·49-s + 0.544·51-s − 0.541·53-s − 0.722·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 - 1.35T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 6.52T + 23T^{2} \) |
| 29 | \( 1 + 2.77T + 29T^{2} \) |
| 31 | \( 1 - 0.550T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 5.97T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 61 | \( 1 + 9.07T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 - 0.357T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.332938717320238982557470733371, −7.47327546562523756608425046178, −6.87190157664936915522302479362, −6.12182532723656472919328187387, −4.91417805265217956227843769188, −3.78326153913444061916626447973, −3.30524496039140590385086795097, −3.18475037469604285800270283005, −1.37792938896441972833407763249, 0,
1.37792938896441972833407763249, 3.18475037469604285800270283005, 3.30524496039140590385086795097, 3.78326153913444061916626447973, 4.91417805265217956227843769188, 6.12182532723656472919328187387, 6.87190157664936915522302479362, 7.47327546562523756608425046178, 8.332938717320238982557470733371
