| L(s) = 1 | − 3·3-s − 5·5-s + 32·7-s + 9·9-s + 60·11-s + 34·13-s + 15·15-s + 42·17-s + 76·19-s − 96·21-s + 25·25-s − 27·27-s − 6·29-s − 232·31-s − 180·33-s − 160·35-s − 134·37-s − 102·39-s + 234·41-s + 412·43-s − 45·45-s − 360·47-s + 681·49-s − 126·51-s − 222·53-s − 300·55-s − 228·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.72·7-s + 1/3·9-s + 1.64·11-s + 0.725·13-s + 0.258·15-s + 0.599·17-s + 0.917·19-s − 0.997·21-s + 1/5·25-s − 0.192·27-s − 0.0384·29-s − 1.34·31-s − 0.949·33-s − 0.772·35-s − 0.595·37-s − 0.418·39-s + 0.891·41-s + 1.46·43-s − 0.149·45-s − 1.11·47-s + 1.98·49-s − 0.345·51-s − 0.575·53-s − 0.735·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.469751729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.469751729\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 + 134 T + p^{3} T^{2} \) |
| 41 | \( 1 - 234 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 360 T + p^{3} T^{2} \) |
| 53 | \( 1 + 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 490 T + p^{3} T^{2} \) |
| 67 | \( 1 + 812 T + p^{3} T^{2} \) |
| 71 | \( 1 - 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 746 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 804 T + p^{3} T^{2} \) |
| 89 | \( 1 + 678 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−9.540050796522078336073436309397, −8.806766553937262050452493064420, −7.87198357468062230403733222641, −7.22787853778483760339091090957, −6.12592618835468093540241336367, −5.25606096269326147510348016141, −4.35511860024375186690438094227, −3.55271980986285978987792811032, −1.68017094596186336177073926417, −0.992117520111480849416600229549,
0.992117520111480849416600229549, 1.68017094596186336177073926417, 3.55271980986285978987792811032, 4.35511860024375186690438094227, 5.25606096269326147510348016141, 6.12592618835468093540241336367, 7.22787853778483760339091090957, 7.87198357468062230403733222641, 8.806766553937262050452493064420, 9.540050796522078336073436309397
