| L(s) = 1 | − 2·4-s + 3·5-s − 3·7-s − 3·9-s − 11-s + 13-s + 4·16-s + 2·17-s − 19-s − 6·20-s + 2·23-s + 4·25-s + 6·28-s + 5·29-s + 4·31-s − 9·35-s + 6·36-s − 9·37-s + 2·41-s − 4·43-s + 2·44-s − 9·45-s + 2·49-s − 2·52-s + 6·53-s − 3·55-s + 10·59-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 1.13·7-s − 9-s − 0.301·11-s + 0.277·13-s + 16-s + 0.485·17-s − 0.229·19-s − 1.34·20-s + 0.417·23-s + 4/5·25-s + 1.13·28-s + 0.928·29-s + 0.718·31-s − 1.52·35-s + 36-s − 1.47·37-s + 0.312·41-s − 0.609·43-s + 0.301·44-s − 1.34·45-s + 2/7·49-s − 0.277·52-s + 0.824·53-s − 0.404·55-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 463 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.918665836298322253919098648305, −6.72866830699259125168528036722, −6.26163725232583107316802310074, −5.44884205695622927284989868155, −5.17572561424908293942273736701, −3.97206916625008675612818390326, −3.13610900247058449984626442153, −2.51326240031036173656539818773, −1.19544752801895269089094788402, 0,
1.19544752801895269089094788402, 2.51326240031036173656539818773, 3.13610900247058449984626442153, 3.97206916625008675612818390326, 5.17572561424908293942273736701, 5.44884205695622927284989868155, 6.26163725232583107316802310074, 6.72866830699259125168528036722, 7.918665836298322253919098648305
