| L(s) = 1 | + (1.28 + 0.588i)2-s + (−0.509 − 1.65i)3-s + (1.30 + 1.51i)4-s + (0.263 + 0.300i)5-s + (0.318 − 2.42i)6-s + (2.25 + 0.297i)7-s + (0.791 + 2.71i)8-s + (−2.48 + 1.68i)9-s + (0.162 + 0.542i)10-s + (2.51 + 5.09i)11-s + (1.83 − 2.93i)12-s + (−0.728 − 2.14i)13-s + (2.72 + 1.71i)14-s + (0.363 − 0.590i)15-s + (−0.578 + 3.95i)16-s + (0.935 + 0.935i)17-s + ⋯ |
| L(s) = 1 | + (0.909 + 0.415i)2-s + (−0.294 − 0.955i)3-s + (0.653 + 0.756i)4-s + (0.117 + 0.134i)5-s + (0.129 − 0.991i)6-s + (0.853 + 0.112i)7-s + (0.279 + 0.960i)8-s + (−0.826 + 0.562i)9-s + (0.0513 + 0.171i)10-s + (0.758 + 1.53i)11-s + (0.530 − 0.847i)12-s + (−0.202 − 0.595i)13-s + (0.729 + 0.457i)14-s + (0.0938 − 0.152i)15-s + (−0.144 + 0.989i)16-s + (0.227 + 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48841 + 0.463988i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48841 + 0.463988i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.28 - 0.588i)T \) |
| 3 | \( 1 + (0.509 + 1.65i)T \) |
| good | 5 | \( 1 + (-0.263 - 0.300i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-2.25 - 0.297i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-2.51 - 5.09i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (0.728 + 2.14i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (-0.935 - 0.935i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.719 + 3.61i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.735 + 5.59i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-8.78 + 0.575i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (3.85 - 2.22i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.930 - 4.67i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.843 + 6.40i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (10.5 - 5.19i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (6.67 - 1.78i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.873 + 1.30i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-2.57 - 2.93i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.219 + 3.34i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (5.54 + 2.73i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-1.44 - 3.48i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-5.52 + 13.3i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.11 - 11.6i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (5.97 + 5.24i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (2.28 - 0.947i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (8.69 + 5.01i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.09696202618648410176929501577, −10.12428899962205173161119551724, −8.541062078155014146560442187855, −7.919904971567657035473386967349, −6.87138086519621187877264744188, −6.46565660847223629638994816780, −5.10809627069998938919880300695, −4.54717825670509167739726268585, −2.80172349153289067646699152679, −1.73326868122605458776345584200,
1.39292704588641196894712131942, 3.22097639920429479936288793668, 3.96838979236499488634265284450, 5.06156977229546851563827609597, 5.70417079773032495014243472870, 6.68470093288042384172032317047, 8.135337133202792732311173031993, 9.180555836170651186116570744954, 9.984112923266399501563357251019, 10.95476098967933951623284198621
