| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 2·5-s + 9·6-s − 21·8-s − 18·9-s + 6·10-s + 11·11-s + 3·12-s − 73·13-s + 6·15-s − 71·16-s + 62·17-s − 54·18-s − 84·19-s + 2·20-s + 33·22-s + 124·23-s − 63·24-s − 121·25-s − 219·26-s − 135·27-s − 203·29-s + 18·30-s − 224·31-s − 45·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.178·5-s + 0.612·6-s − 0.928·8-s − 2/3·9-s + 0.189·10-s + 0.301·11-s + 0.0721·12-s − 1.55·13-s + 0.103·15-s − 1.10·16-s + 0.884·17-s − 0.707·18-s − 1.01·19-s + 0.0223·20-s + 0.319·22-s + 1.12·23-s − 0.535·24-s − 0.967·25-s − 1.65·26-s − 0.962·27-s − 1.29·29-s + 0.109·30-s − 1.29·31-s − 0.248·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - p T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 13 | \( 1 + 73 T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 + 84 T + p^{3} T^{2} \) |
| 23 | \( 1 - 124 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 224 T + p^{3} T^{2} \) |
| 37 | \( 1 - 412 T + p^{3} T^{2} \) |
| 41 | \( 1 + 176 T + p^{3} T^{2} \) |
| 43 | \( 1 - 400 T + p^{3} T^{2} \) |
| 47 | \( 1 + 586 T + p^{3} T^{2} \) |
| 53 | \( 1 + 234 T + p^{3} T^{2} \) |
| 59 | \( 1 - 9 p T + p^{3} T^{2} \) |
| 61 | \( 1 - 367 T + p^{3} T^{2} \) |
| 67 | \( 1 - 105 T + p^{3} T^{2} \) |
| 71 | \( 1 + 878 T + p^{3} T^{2} \) |
| 73 | \( 1 + 236 T + p^{3} T^{2} \) |
| 79 | \( 1 - 351 T + p^{3} T^{2} \) |
| 83 | \( 1 + 342 T + p^{3} T^{2} \) |
| 89 | \( 1 - 366 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1001 T + p^{3} T^{2} \) |
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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
−9.683676141001956187553820881261, −9.257023987640682236840111945280, −8.158950582235844145132144534358, −7.18741480473236826520931941448, −5.94193576193454733088078320896, −5.23898679958221832449770463872, −4.15691274875767352179008112046, −3.16299565262697374025410037027, −2.20656561849245156673705732316, 0,
2.20656561849245156673705732316, 3.16299565262697374025410037027, 4.15691274875767352179008112046, 5.23898679958221832449770463872, 5.94193576193454733088078320896, 7.18741480473236826520931941448, 8.158950582235844145132144534358, 9.257023987640682236840111945280, 9.683676141001956187553820881261
