| L(s) = 1 | + 5·5-s + 30·7-s + 50·11-s − 20·13-s + 10·17-s + 44·19-s + 120·23-s + 25·25-s + 50·29-s − 108·31-s + 150·35-s − 40·37-s − 400·41-s − 280·43-s − 280·47-s + 557·49-s + 610·53-s + 250·55-s + 50·59-s − 518·61-s − 100·65-s + 180·67-s + 700·71-s − 410·73-s + 1.50e3·77-s + 516·79-s + 660·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.61·7-s + 1.37·11-s − 0.426·13-s + 0.142·17-s + 0.531·19-s + 1.08·23-s + 1/5·25-s + 0.320·29-s − 0.625·31-s + 0.724·35-s − 0.177·37-s − 1.52·41-s − 0.993·43-s − 0.868·47-s + 1.62·49-s + 1.58·53-s + 0.612·55-s + 0.110·59-s − 1.08·61-s − 0.190·65-s + 0.328·67-s + 1.17·71-s − 0.657·73-s + 2.22·77-s + 0.734·79-s + 0.872·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.065613366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.065613366\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 10 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 + 400 T + p^{3} T^{2} \) |
| 43 | \( 1 + 280 T + p^{3} T^{2} \) |
| 47 | \( 1 + 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 610 T + p^{3} T^{2} \) |
| 59 | \( 1 - 50 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 - 516 T + p^{3} T^{2} \) |
| 83 | \( 1 - 660 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1500 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1630 T + p^{3} T^{2} \) |
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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
−9.993291932847492572798688798393, −9.074024057513124365500201026682, −8.400454716198910678421739467466, −7.38090143772431656463624801256, −6.56993996054172596977646320094, −5.31683199692416570189910112914, −4.72035791292709116070499936319, −3.48110272126592208948751840479, −1.96944074776692490227011829391, −1.12627939473566469899123802418,
1.12627939473566469899123802418, 1.96944074776692490227011829391, 3.48110272126592208948751840479, 4.72035791292709116070499936319, 5.31683199692416570189910112914, 6.56993996054172596977646320094, 7.38090143772431656463624801256, 8.400454716198910678421739467466, 9.074024057513124365500201026682, 9.993291932847492572798688798393
