| L(s) = 1 | + (−1.30 + 1.83i)3-s + (−3.28 + 0.632i)5-s + (−2.64 − 0.0411i)7-s + (−0.672 − 1.94i)9-s + (0.952 − 0.748i)11-s + (−1.12 + 0.720i)13-s + (3.12 − 6.83i)15-s + (1.56 + 1.49i)17-s + (1.11 − 1.06i)19-s + (3.52 − 4.79i)21-s + (2.24 − 4.23i)23-s + (5.72 − 2.29i)25-s + (−2.03 − 0.598i)27-s + (1.76 − 0.519i)29-s + (−1.02 + 0.0983i)31-s + ⋯ |
| L(s) = 1 | + (−0.752 + 1.05i)3-s + (−1.46 + 0.282i)5-s + (−0.999 − 0.0155i)7-s + (−0.224 − 0.647i)9-s + (0.287 − 0.225i)11-s + (−0.311 + 0.199i)13-s + (0.805 − 1.76i)15-s + (0.380 + 0.363i)17-s + (0.256 − 0.244i)19-s + (0.769 − 1.04i)21-s + (0.468 − 0.883i)23-s + (1.14 − 0.458i)25-s + (−0.392 − 0.115i)27-s + (0.328 − 0.0965i)29-s + (−0.184 + 0.0176i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.260591 - 0.138094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.260591 - 0.138094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0411i)T \) |
| 23 | \( 1 + (-2.24 + 4.23i)T \) |
| good | 3 | \( 1 + (1.30 - 1.83i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (3.28 - 0.632i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-0.952 + 0.748i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.12 - 0.720i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 1.49i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.11 + 1.06i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-1.76 + 0.519i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.02 - 0.0983i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (1.24 + 3.60i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-4.84 - 5.59i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.83 + 4.00i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.27 + 7.40i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.387 - 8.13i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (6.78 + 3.49i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (6.10 + 8.57i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (7.64 - 3.05i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-1.93 + 13.4i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.531 - 2.19i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (0.351 + 7.37i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (10.4 - 12.1i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.61 - 0.345i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (11.1 + 12.8i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61719320638486765372985615461, −9.709159113518046731049735570684, −8.854990762214518231946018585977, −7.72369833695689351875469862369, −6.83583070055723921985730935341, −5.85709354611425825422104155064, −4.66385107593573604253078941258, −3.93213134610465823465762242458, −3.06263960185681505708193584362, −0.21700711040539659695904713397,
1.07405776773550922173861971235, 3.05353275393328416047150776932, 4.10840733952153482100189821992, 5.37943935284120331130198116028, 6.38135250741581445144491842191, 7.30782998227560794369673544265, 7.64992472373915974925427184976, 8.884739397134682834158895478945, 9.834266197719783188164670408101, 11.00878590607320986733101385145
