| L(s) = 1 | + (−0.723 − 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (1.80 − 1.42i)5-s + (−0.841 − 0.540i)6-s + (1.41 + 2.23i)7-s + (0.654 − 0.755i)8-s + (0.928 − 0.371i)9-s + (−2.29 − 0.218i)10-s + (4.30 − 4.10i)11-s + (0.235 + 0.971i)12-s + (−2.24 − 4.90i)13-s + (0.513 − 2.59i)14-s + (1.50 − 1.73i)15-s + (−0.995 + 0.0950i)16-s + (−3.78 + 1.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.511 − 0.487i)2-s + (0.566 − 0.109i)3-s + (0.0237 + 0.499i)4-s + (0.809 − 0.636i)5-s + (−0.343 − 0.220i)6-s + (0.536 + 0.843i)7-s + (0.231 − 0.267i)8-s + (0.309 − 0.123i)9-s + (−0.724 − 0.0691i)10-s + (1.29 − 1.23i)11-s + (0.0680 + 0.280i)12-s + (−0.621 − 1.36i)13-s + (0.137 − 0.693i)14-s + (0.389 − 0.449i)15-s + (−0.248 + 0.0237i)16-s + (−0.918 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50581 - 1.08923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50581 - 1.08923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.723 + 0.690i)T \) |
| 3 | \( 1 + (-0.981 + 0.189i)T \) |
| 7 | \( 1 + (-1.41 - 2.23i)T \) |
| 23 | \( 1 + (-2.89 - 3.82i)T \) |
| good | 5 | \( 1 + (-1.80 + 1.42i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (-4.30 + 4.10i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.24 + 4.90i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (3.78 - 1.95i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (3.36 + 1.73i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (-5.14 - 3.30i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.03 + 5.87i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (-5.22 + 2.09i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (1.02 + 7.14i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.64 - 6.51i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-4.50 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.22 + 5.93i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (6.93 + 0.662i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (2.98 + 0.575i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (-0.549 + 2.26i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (6.24 - 1.83i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.328 - 6.89i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (-9.29 + 13.0i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (0.203 - 1.41i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (3.74 - 10.8i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-0.428 - 2.98i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−9.448386451513863687546364944388, −9.118021556528413635941519107432, −8.485309539439075786759590177051, −7.69873690280645359077236667657, −6.35103088939541123696224494479, −5.58614543716812966113255866810, −4.43710424465035788467075218334, −3.15265671977053646070775028444, −2.18173693636801971402475546252, −1.06827172893096936789282138143,
1.60012940671088376136297942277, 2.41966710419234755659935107460, 4.27436636121118329024822397977, 4.63933366092089560122436941887, 6.47552980672195749276152449328, 6.75878165215717826673436441842, 7.49943189614996333259744631792, 8.709880539007023060946641709858, 9.306916962502536160391977411197, 10.03039653028806236867846229354
