| L(s) = 1 | + (0.984 − 0.173i)2-s + (1.62 + 0.598i)3-s + (0.939 − 0.342i)4-s + (1.32 + 1.80i)5-s + (1.70 + 0.307i)6-s + (−0.260 − 0.150i)7-s + (0.866 − 0.5i)8-s + (2.28 + 1.94i)9-s + (1.61 + 1.54i)10-s + (−1.17 + 0.680i)11-s + (1.73 + 0.00677i)12-s + (−2.32 − 1.94i)13-s + (−0.282 − 0.102i)14-s + (1.06 + 3.72i)15-s + (0.766 − 0.642i)16-s + (−0.0110 − 0.0629i)17-s + ⋯ |
| L(s) = 1 | + (0.696 − 0.122i)2-s + (0.938 + 0.345i)3-s + (0.469 − 0.171i)4-s + (0.590 + 0.806i)5-s + (0.695 + 0.125i)6-s + (−0.0985 − 0.0569i)7-s + (0.306 − 0.176i)8-s + (0.760 + 0.648i)9-s + (0.510 + 0.489i)10-s + (−0.355 + 0.205i)11-s + (0.499 + 0.00195i)12-s + (−0.644 − 0.540i)13-s + (−0.0756 − 0.0275i)14-s + (0.275 + 0.961i)15-s + (0.191 − 0.160i)16-s + (−0.00269 − 0.0152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.00191 + 0.730871i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.00191 + 0.730871i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.62 - 0.598i)T \) |
| 5 | \( 1 + (-1.32 - 1.80i)T \) |
| 19 | \( 1 + (3.34 + 2.79i)T \) |
| good | 7 | \( 1 + (0.260 + 0.150i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.17 - 0.680i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.32 + 1.94i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0110 + 0.0629i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.311 - 0.113i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 7.39i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.02 + 1.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 + (-4.00 + 3.35i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.86 - 5.13i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.167 + 0.949i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.00 - 5.49i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.12 + 6.40i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.91 + 1.06i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.99 - 11.3i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.92 - 2.88i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (6.48 + 7.72i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (9.29 + 11.0i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.598 - 1.03i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.2 - 8.62i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.21 + 6.90i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.62315901200533528365103503998, −10.05477739872237548414872861817, −9.271690058584741376564588843724, −8.000327154433790837574178940332, −7.21951831609746168753261222921, −6.22017866913786949674875804184, −5.07763779078657857371935880772, −4.02943836917865180830257712019, −2.85451019744644352033967626211, −2.19773038922301141192371446990,
1.64664726562059179356081321933, 2.72319430483487381648534974043, 4.01685717500351871787851011433, 4.98610490334792958593613218967, 6.09012782555427291451503468101, 7.03076041496803957246547744180, 8.049094579919907769941539821781, 8.828498921339623017060775955768, 9.654903637006341451372586543603, 10.58264792223209095223682183771
