L(s) = 1 | + (2.82 + 4.89i)2-s + (−36.7 − 21.2i)3-s + (−15.9 + 27.7i)4-s + (−162. + 93.7i)5-s − 239. i·6-s + (141. − 312. i)7-s − 181.·8-s + (535. + 927. i)9-s + (−918. − 530. i)10-s + (−555. + 962. i)11-s + (1.17e3 − 678. i)12-s + 706. i·13-s + (1.93e3 − 192. i)14-s + 7.95e3·15-s + (−512. − 886. i)16-s + (−7.33e3 − 4.23e3i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−1.36 − 0.785i)3-s + (−0.249 + 0.433i)4-s + (−1.29 + 0.749i)5-s − 1.11i·6-s + (0.411 − 0.911i)7-s − 0.353·8-s + (0.734 + 1.27i)9-s + (−0.918 − 0.530i)10-s + (−0.417 + 0.723i)11-s + (0.680 − 0.392i)12-s + 0.321i·13-s + (0.703 − 0.0700i)14-s + 2.35·15-s + (−0.125 − 0.216i)16-s + (−1.49 − 0.861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.00921571 - 0.0812319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00921571 - 0.0812319i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 - 4.89i)T \) |
| 7 | \( 1 + (-141. + 312. i)T \) |
good | 3 | \( 1 + (36.7 + 21.2i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (162. - 93.7i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (555. - 962. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 706. iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (7.33e3 + 4.23e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-22.0 + 12.7i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-424. - 735. i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 1.51e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (1.20e3 + 696. i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-4.64e3 - 8.04e3i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 1.09e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 4.55e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.19e5 + 6.92e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (3.09e4 - 5.35e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-6.80e4 - 3.92e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.51e5 - 8.73e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.17e5 + 3.77e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.61e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.92e5 + 1.10e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.33e5 + 5.77e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 6.53e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.14e5 + 6.61e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.59e6iT - 8.32e11T^{2} \) |
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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
−18.52586266136478741265667057117, −17.63827577863054274051413971691, −16.37562777080321969994998800084, −15.09860309205796285495790031491, −13.35317877876274823530668912210, −11.83950897215917063279934054986, −10.95286841991577160020943939037, −7.55035587508063884003396003026, −6.79581831359207248185149357452, −4.56593239874600100182625789833,
0.06301963162225269464454508655, 4.26718284075360051888313407319, 5.58322959317473051261828330436, 8.676689252786952678461792689477, 10.83530825046464348688806416065, 11.63140187449323091173064775614, 12.70830630250047383097237723385, 15.30135836739645629999603408427, 15.99465338923401800473278822160, 17.47885838666297571642242453019