| L(s) = 1 | + (0.782 + 1.17i)2-s + (0.703 − 0.176i)3-s + (−0.775 + 1.84i)4-s + (1.02 + 0.484i)5-s + (0.758 + 0.691i)6-s + (0.442 + 1.45i)7-s + (−2.77 + 0.529i)8-s + (−2.18 + 1.16i)9-s + (0.230 + 1.58i)10-s + (1.21 − 0.180i)11-s + (−0.220 + 1.43i)12-s + (1.16 − 3.26i)13-s + (−1.37 + 1.66i)14-s + (0.805 + 0.160i)15-s + (−2.79 − 2.85i)16-s + (4.99 − 0.992i)17-s + ⋯ |
| L(s) = 1 | + (0.553 + 0.832i)2-s + (0.406 − 0.101i)3-s + (−0.387 + 0.921i)4-s + (0.457 + 0.216i)5-s + (0.309 + 0.282i)6-s + (0.167 + 0.551i)7-s + (−0.982 + 0.187i)8-s + (−0.727 + 0.388i)9-s + (0.0729 + 0.501i)10-s + (0.367 − 0.0544i)11-s + (−0.0636 + 0.414i)12-s + (0.323 − 0.904i)13-s + (−0.366 + 0.444i)14-s + (0.208 + 0.0413i)15-s + (−0.699 − 0.714i)16-s + (1.21 − 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00354 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00354 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30038 + 1.30500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30038 + 1.30500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.782 - 1.17i)T \) |
| good | 3 | \( 1 + (-0.703 + 0.176i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 0.484i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-0.442 - 1.45i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 0.180i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-1.16 + 3.26i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-4.99 + 0.992i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (-2.64 - 2.39i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (3.37 - 0.332i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (1.72 + 2.33i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (0.335 + 0.139i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.167 + 3.40i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-6.53 + 7.96i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (-0.966 + 3.85i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (4.45 + 2.97i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (3.06 - 4.13i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-3.24 - 9.06i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (2.64 - 4.41i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-4.77 - 2.86i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (6.09 - 11.4i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-1.95 + 6.43i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (4.98 + 7.45i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.118 - 2.41i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (12.9 + 1.27i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-4.14 + 10.0i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.36027505964547534355751011598, −11.64119217709569991412324887204, −10.21326946535743762121561439015, −9.081889221620225569612844466547, −8.165501262220589795283401210208, −7.43965811542451705422748195321, −5.80194532152777536427353273914, −5.58776879971616547963199277239, −3.76057528192532510628879441163, −2.56881151242157261314670591855,
1.47604284364554123682461886672, 3.10931995793559165785077709515, 4.15774174108555796100313668555, 5.46300125979323823157142840665, 6.47781272682768976857993365024, 8.060081747726469883724465919927, 9.343252864372002804395672496262, 9.714355263078128738332427748999, 11.06667516871869033354652306405, 11.70306380050116534369018168096
