| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s + 5·13-s + 14-s − 16-s + 7·17-s + 6·19-s − 4·23-s + 5·26-s − 28-s + 9·29-s − 2·31-s + 5·32-s + 7·34-s − 9·37-s + 6·38-s − 7·41-s − 6·43-s − 4·46-s + 2·47-s + 49-s − 5·52-s + 3·53-s − 3·56-s + 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.69·17-s + 1.37·19-s − 0.834·23-s + 0.980·26-s − 0.188·28-s + 1.67·29-s − 0.359·31-s + 0.883·32-s + 1.20·34-s − 1.47·37-s + 0.973·38-s − 1.09·41-s − 0.914·43-s − 0.589·46-s + 0.291·47-s + 1/7·49-s − 0.693·52-s + 0.412·53-s − 0.400·56-s + 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.613398065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.613398065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.34042942886786, −12.55433646084485, −12.15090394263221, −11.84115082009456, −11.54959000064383, −10.65756646992484, −10.22976619704403, −9.979509474554720, −9.219578366192154, −8.819264048900862, −8.335459010954945, −7.921475041219509, −7.402890313856595, −6.669055772193624, −6.160871515553352, −5.694299766401794, −5.197185203825361, −4.904967052453859, −4.119770363581002, −3.635084724141832, −3.229909500420843, −2.815980287268511, −1.633642602079264, −1.283907932670471, −0.5166979766345698,
0.5166979766345698, 1.283907932670471, 1.633642602079264, 2.815980287268511, 3.229909500420843, 3.635084724141832, 4.119770363581002, 4.904967052453859, 5.197185203825361, 5.694299766401794, 6.160871515553352, 6.669055772193624, 7.402890313856595, 7.921475041219509, 8.335459010954945, 8.819264048900862, 9.219578366192154, 9.979509474554720, 10.22976619704403, 10.65756646992484, 11.54959000064383, 11.84115082009456, 12.15090394263221, 12.55433646084485, 13.34042942886786
