| L(s) = 1 | + 5·5-s + 22·7-s − 12·11-s + 38·13-s + 105·17-s + 157·19-s − 117·23-s + 25·25-s − 66·29-s + 25·31-s + 110·35-s + 314·37-s + 504·41-s − 380·43-s − 252·47-s + 141·49-s − 3·53-s − 60·55-s − 318·59-s + 293·61-s + 190·65-s + 322·67-s − 120·71-s + 44·73-s − 264·77-s − 917·79-s + 309·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.18·7-s − 0.328·11-s + 0.810·13-s + 1.49·17-s + 1.89·19-s − 1.06·23-s + 1/5·25-s − 0.422·29-s + 0.144·31-s + 0.531·35-s + 1.39·37-s + 1.91·41-s − 1.34·43-s − 0.782·47-s + 0.411·49-s − 0.00777·53-s − 0.147·55-s − 0.701·59-s + 0.614·61-s + 0.362·65-s + 0.587·67-s − 0.200·71-s + 0.0705·73-s − 0.390·77-s − 1.30·79-s + 0.408·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.535899116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.535899116\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 105 T + p^{3} T^{2} \) |
| 19 | \( 1 - 157 T + p^{3} T^{2} \) |
| 23 | \( 1 + 117 T + p^{3} T^{2} \) |
| 29 | \( 1 + 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 25 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 504 T + p^{3} T^{2} \) |
| 43 | \( 1 + 380 T + p^{3} T^{2} \) |
| 47 | \( 1 + 252 T + p^{3} T^{2} \) |
| 53 | \( 1 + 3 T + p^{3} T^{2} \) |
| 59 | \( 1 + 318 T + p^{3} T^{2} \) |
| 61 | \( 1 - 293 T + p^{3} T^{2} \) |
| 67 | \( 1 - 322 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 44 T + p^{3} T^{2} \) |
| 79 | \( 1 + 917 T + p^{3} T^{2} \) |
| 83 | \( 1 - 309 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1328 T + p^{3} T^{2} \) |
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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
−8.600224127229223504736382379721, −7.82717285630202922929664470569, −7.50239371313490625592441508240, −6.14223266970085526524793455680, −5.55887653795517989118154070915, −4.86109668270508664550614353224, −3.78582226702215965501330683334, −2.84500460654417303921612111355, −1.63496497663678019117224170964, −0.944946556762391867027290063342,
0.944946556762391867027290063342, 1.63496497663678019117224170964, 2.84500460654417303921612111355, 3.78582226702215965501330683334, 4.86109668270508664550614353224, 5.55887653795517989118154070915, 6.14223266970085526524793455680, 7.50239371313490625592441508240, 7.82717285630202922929664470569, 8.600224127229223504736382379721
