| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 6·6-s + 8·7-s + 8·8-s + 9·9-s − 38·11-s − 12·12-s + 13·13-s + 16·14-s + 16·16-s + 78·17-s + 18·18-s − 72·19-s − 24·21-s − 76·22-s + 52·23-s − 24·24-s + 26·26-s − 27·27-s + 32·28-s + 242·29-s + 76·31-s + 32·32-s + 114·33-s + 156·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.431·7-s + 0.353·8-s + 1/3·9-s − 1.04·11-s − 0.288·12-s + 0.277·13-s + 0.305·14-s + 1/4·16-s + 1.11·17-s + 0.235·18-s − 0.869·19-s − 0.249·21-s − 0.736·22-s + 0.471·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.215·28-s + 1.54·29-s + 0.440·31-s + 0.176·32-s + 0.601·33-s + 0.786·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.918825787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918825787\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 72 T + p^{3} T^{2} \) |
| 23 | \( 1 - 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 - 76 T + p^{3} T^{2} \) |
| 37 | \( 1 + 342 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 - 854 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 838 T + p^{3} T^{2} \) |
| 73 | \( 1 - 970 T + p^{3} T^{2} \) |
| 79 | \( 1 + 352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 474 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1452 T + p^{3} T^{2} \) |
| 97 | \( 1 - 562 T + p^{3} T^{2} \) |
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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
−8.521575426776673833551044377633, −8.083892026757563840069490985831, −7.00291295455377976315977720218, −6.43015615924418912398270848029, −5.23718714409103766927614542348, −5.13934718853336050456713627913, −3.96886996505258410181297524527, −3.01021187157966889470325559163, −1.92800305311966110219632383644, −0.73450885588838313224601393872,
0.73450885588838313224601393872, 1.92800305311966110219632383644, 3.01021187157966889470325559163, 3.96886996505258410181297524527, 5.13934718853336050456713627913, 5.23718714409103766927614542348, 6.43015615924418912398270848029, 7.00291295455377976315977720218, 8.083892026757563840069490985831, 8.521575426776673833551044377633
