| L(s) = 1 | + 8·5-s + 10·7-s + 68·11-s + 46·13-s + 74·17-s − 16·19-s − 20·23-s − 61·25-s + 228·29-s + 162·31-s + 80·35-s − 262·37-s − 30·41-s − 264·43-s + 124·47-s − 243·49-s − 204·53-s + 544·55-s + 340·59-s − 950·61-s + 368·65-s + 436·67-s − 780·71-s + 518·73-s + 680·77-s + 1.01e3·79-s + 852·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s + 0.539·7-s + 1.86·11-s + 0.981·13-s + 1.05·17-s − 0.193·19-s − 0.181·23-s − 0.487·25-s + 1.45·29-s + 0.938·31-s + 0.386·35-s − 1.16·37-s − 0.114·41-s − 0.936·43-s + 0.384·47-s − 0.708·49-s − 0.528·53-s + 1.33·55-s + 0.750·59-s − 1.99·61-s + 0.702·65-s + 0.795·67-s − 1.30·71-s + 0.830·73-s + 1.00·77-s + 1.43·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.410738330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.410738330\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 68 T + p^{3} T^{2} \) |
| 13 | \( 1 - 46 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 20 T + p^{3} T^{2} \) |
| 29 | \( 1 - 228 T + p^{3} T^{2} \) |
| 31 | \( 1 - 162 T + p^{3} T^{2} \) |
| 37 | \( 1 + 262 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 264 T + p^{3} T^{2} \) |
| 47 | \( 1 - 124 T + p^{3} T^{2} \) |
| 53 | \( 1 + 204 T + p^{3} T^{2} \) |
| 59 | \( 1 - 340 T + p^{3} T^{2} \) |
| 61 | \( 1 + 950 T + p^{3} T^{2} \) |
| 67 | \( 1 - 436 T + p^{3} T^{2} \) |
| 71 | \( 1 + 780 T + p^{3} T^{2} \) |
| 73 | \( 1 - 518 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 83 | \( 1 - 852 T + p^{3} T^{2} \) |
| 89 | \( 1 - 686 T + p^{3} T^{2} \) |
| 97 | \( 1 + 806 T + p^{3} T^{2} \) |
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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
−9.427967877865663366980303005227, −8.637277690690378348429541627206, −7.939113070363320755289354577086, −6.61807292581900610560912487618, −6.24831832231445004776805467695, −5.19225755927240523488252519206, −4.15211738156890261060044485256, −3.25273285579526135422668884320, −1.75190653832476085050999868640, −1.08854690164184933977417403689,
1.08854690164184933977417403689, 1.75190653832476085050999868640, 3.25273285579526135422668884320, 4.15211738156890261060044485256, 5.19225755927240523488252519206, 6.24831832231445004776805467695, 6.61807292581900610560912487618, 7.939113070363320755289354577086, 8.637277690690378348429541627206, 9.427967877865663366980303005227
