| L(s) = 1 | − 3-s − 2·4-s + 9-s + 2·12-s + 13-s + 4·16-s + 7·17-s − 8·19-s + 7·23-s − 27-s − 6·29-s − 8·31-s − 2·36-s − 6·37-s − 39-s − 41-s − 12·43-s − 6·47-s − 4·48-s − 7·51-s − 2·52-s − 13·53-s + 8·57-s + 7·59-s − 11·61-s − 8·64-s + 7·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s + 0.277·13-s + 16-s + 1.69·17-s − 1.83·19-s + 1.45·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 1/3·36-s − 0.986·37-s − 0.160·39-s − 0.156·41-s − 1.82·43-s − 0.875·47-s − 0.577·48-s − 0.980·51-s − 0.277·52-s − 1.78·53-s + 1.05·57-s + 0.911·59-s − 1.40·61-s − 64-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5770667249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5770667249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.65634971935704, −14.11157320432445, −13.41556376571103, −12.89786836739907, −12.67077048055248, −12.19452071277416, −11.33268412365592, −11.00530439075174, −10.38294559887354, −9.941139539369242, −9.316711777526075, −8.899833740932385, −8.215413176675912, −7.846853208224313, −7.048652451186459, −6.536080651715937, −5.845341444396875, −5.184410570658617, −5.036350338063573, −4.157174544942835, −3.549283516018833, −3.151953416513570, −1.848658544524728, −1.340449752576249, −0.2974357917524894,
0.2974357917524894, 1.340449752576249, 1.848658544524728, 3.151953416513570, 3.549283516018833, 4.157174544942835, 5.036350338063573, 5.184410570658617, 5.845341444396875, 6.536080651715937, 7.048652451186459, 7.846853208224313, 8.215413176675912, 8.899833740932385, 9.316711777526075, 9.941139539369242, 10.38294559887354, 11.00530439075174, 11.33268412365592, 12.19452071277416, 12.67077048055248, 12.89786836739907, 13.41556376571103, 14.11157320432445, 14.65634971935704
