| L(s) = 1 | + 5-s + 4·7-s + 6·11-s − 4·13-s − 6·17-s + 6·23-s + 25-s − 2·29-s + 4·35-s − 8·37-s − 10·41-s + 4·43-s − 2·47-s + 9·49-s − 10·53-s + 6·55-s − 12·59-s + 2·61-s − 4·65-s + 8·67-s − 6·73-s + 24·77-s − 4·79-s + 2·83-s − 6·85-s + 14·89-s − 16·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.80·11-s − 1.10·13-s − 1.45·17-s + 1.25·23-s + 1/5·25-s − 0.371·29-s + 0.676·35-s − 1.31·37-s − 1.56·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s − 1.37·53-s + 0.809·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 0.977·67-s − 0.702·73-s + 2.73·77-s − 0.450·79-s + 0.219·83-s − 0.650·85-s + 1.48·89-s − 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.451237340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.451237340\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.79294437928752, −12.28801705518973, −11.77383113663903, −11.44266973378123, −11.13274765809583, −10.51618907549901, −10.15210018711814, −9.277331067661983, −9.151153011816048, −8.796413461360836, −8.186841295058787, −7.620196025064141, −7.115840963090403, −6.623306785020152, −6.391615946139337, −5.514256164149810, −5.015815146088849, −4.685767466127527, −4.278694763085302, −3.558981967253657, −2.972384200626922, −2.135292594239468, −1.758170236812473, −1.377149553315182, −0.4885073943012223,
0.4885073943012223, 1.377149553315182, 1.758170236812473, 2.135292594239468, 2.972384200626922, 3.558981967253657, 4.278694763085302, 4.685767466127527, 5.015815146088849, 5.514256164149810, 6.391615946139337, 6.623306785020152, 7.115840963090403, 7.620196025064141, 8.186841295058787, 8.796413461360836, 9.151153011816048, 9.277331067661983, 10.15210018711814, 10.51618907549901, 11.13274765809583, 11.44266973378123, 11.77383113663903, 12.28801705518973, 12.79294437928752
