| L(s) = 1 | + 10·5-s − 7·7-s + 52·11-s − 10·13-s + 54·17-s − 52·19-s − 48·23-s − 25·25-s + 186·29-s + 224·31-s − 70·35-s + 94·37-s + 478·41-s − 316·43-s − 256·47-s + 49·49-s + 66·53-s + 520·55-s − 420·59-s + 342·61-s − 100·65-s + 668·67-s + 272·71-s − 86·73-s − 364·77-s + 1.36e3·79-s − 188·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 1.42·11-s − 0.213·13-s + 0.770·17-s − 0.627·19-s − 0.435·23-s − 1/5·25-s + 1.19·29-s + 1.29·31-s − 0.338·35-s + 0.417·37-s + 1.82·41-s − 1.12·43-s − 0.794·47-s + 1/7·49-s + 0.171·53-s + 1.27·55-s − 0.926·59-s + 0.717·61-s − 0.190·65-s + 1.21·67-s + 0.454·71-s − 0.137·73-s − 0.538·77-s + 1.93·79-s − 0.248·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.445000052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.445000052\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 - 478 T + p^{3} T^{2} \) |
| 43 | \( 1 + 316 T + p^{3} T^{2} \) |
| 47 | \( 1 + 256 T + p^{3} T^{2} \) |
| 53 | \( 1 - 66 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 - 342 T + p^{3} T^{2} \) |
| 67 | \( 1 - 668 T + p^{3} T^{2} \) |
| 71 | \( 1 - 272 T + p^{3} T^{2} \) |
| 73 | \( 1 + 86 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1360 T + p^{3} T^{2} \) |
| 83 | \( 1 + 188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 366 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1554 T + p^{3} T^{2} \) |
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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
−10.22555601632337802584482656820, −9.725912121458324988973625823637, −8.888254582974282492841881137745, −7.84242898719616973448261780065, −6.49126937440330880576332035648, −6.15527681740539605517095870085, −4.80979631318851918286516580504, −3.65813922421661409365152967147, −2.32445008948095475923961432746, −1.02908229163819369523860264932,
1.02908229163819369523860264932, 2.32445008948095475923961432746, 3.65813922421661409365152967147, 4.80979631318851918286516580504, 6.15527681740539605517095870085, 6.49126937440330880576332035648, 7.84242898719616973448261780065, 8.888254582974282492841881137745, 9.725912121458324988973625823637, 10.22555601632337802584482656820
