| L(s) = 1 | + 25·5-s − 44·7-s + 216·11-s + 770·13-s − 534·17-s − 1.58e3·19-s + 2.90e3·23-s + 625·25-s + 4.56e3·29-s − 2.74e3·31-s − 1.10e3·35-s + 1.44e3·37-s + 1.33e4·41-s − 1.72e4·43-s − 1.08e4·47-s − 1.48e4·49-s + 9.94e3·53-s + 5.40e3·55-s − 1.55e4·59-s + 3.93e4·61-s + 1.92e4·65-s − 5.57e4·67-s + 5.71e4·71-s + 5.04e4·73-s − 9.50e3·77-s + 1.05e4·79-s + 1.08e5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.339·7-s + 0.538·11-s + 1.26·13-s − 0.448·17-s − 1.00·19-s + 1.14·23-s + 1/5·25-s + 1.00·29-s − 0.512·31-s − 0.151·35-s + 0.173·37-s + 1.24·41-s − 1.41·43-s − 0.714·47-s − 0.884·49-s + 0.486·53-s + 0.240·55-s − 0.582·59-s + 1.35·61-s + 0.565·65-s − 1.51·67-s + 1.34·71-s + 1.10·73-s − 0.182·77-s + 0.190·79-s + 1.72·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.536842882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.536842882\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 44 T + p^{5} T^{2} \) |
| 11 | \( 1 - 216 T + p^{5} T^{2} \) |
| 13 | \( 1 - 770 T + p^{5} T^{2} \) |
| 17 | \( 1 + 534 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1580 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2904 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4566 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2744 T + p^{5} T^{2} \) |
| 37 | \( 1 - 1442 T + p^{5} T^{2} \) |
| 41 | \( 1 - 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17204 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10824 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9942 T + p^{5} T^{2} \) |
| 59 | \( 1 + 264 p T + p^{5} T^{2} \) |
| 61 | \( 1 - 39302 T + p^{5} T^{2} \) |
| 67 | \( 1 + 55796 T + p^{5} T^{2} \) |
| 71 | \( 1 - 57120 T + p^{5} T^{2} \) |
| 73 | \( 1 - 50402 T + p^{5} T^{2} \) |
| 79 | \( 1 - 10552 T + p^{5} T^{2} \) |
| 83 | \( 1 - 1308 p T + p^{5} T^{2} \) |
| 89 | \( 1 - 116430 T + p^{5} T^{2} \) |
| 97 | \( 1 + 2782 T + p^{5} T^{2} \) |
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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
−9.523959497780097829881827251852, −8.869218420063491279146969585731, −8.086307272598474803742766428821, −6.65979945634613376979375500395, −6.38063030268993996708743414179, −5.17656513811444323048911936174, −4.09874719434547732630695199075, −3.09991192551333667951075171765, −1.85907281962358327931029873189, −0.76237908923959364563494295161,
0.76237908923959364563494295161, 1.85907281962358327931029873189, 3.09991192551333667951075171765, 4.09874719434547732630695199075, 5.17656513811444323048911936174, 6.38063030268993996708743414179, 6.65979945634613376979375500395, 8.086307272598474803742766428821, 8.869218420063491279146969585731, 9.523959497780097829881827251852
