| L(s) = 1 | + 6.70·2-s + (−7.23 − 12.5i)3-s + 12.9·4-s + (−40.0 − 69.3i)5-s + (−48.5 − 84.0i)6-s + (−74.8 + 129. i)7-s − 127.·8-s + (16.7 − 28.9i)9-s + (−268. − 464. i)10-s + 126.·11-s + (−93.4 − 161. i)12-s + (420. − 729. i)13-s + (−501. + 868. i)14-s + (−579. + 1.00e3i)15-s − 1.27e3·16-s + (1.12e3 − 1.94e3i)17-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + (−0.464 − 0.804i)3-s + 0.403·4-s + (−0.716 − 1.24i)5-s + (−0.550 − 0.952i)6-s + (−0.577 + 0.999i)7-s − 0.706·8-s + (0.0687 − 0.119i)9-s + (−0.848 − 1.46i)10-s + 0.315·11-s + (−0.187 − 0.324i)12-s + (0.690 − 1.19i)13-s + (−0.683 + 1.18i)14-s + (−0.665 + 1.15i)15-s − 1.24·16-s + (0.943 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.611983 - 1.48783i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.611983 - 1.48783i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.21e4 + 462. i)T \) |
| good | 2 | \( 1 - 6.70T + 32T^{2} \) |
| 3 | \( 1 + (7.23 + 12.5i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (40.0 + 69.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (74.8 - 129. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 - 126.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-420. + 729. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-1.12e3 + 1.94e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.46e3 - 2.53e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.99e3 + 3.45e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.13e3 - 1.97e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.14e3 - 1.97e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.25e3 - 2.17e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 192.T + 1.15e8T^{2} \) |
| 47 | \( 1 + 6.42e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.13e4 + 1.96e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + 1.73e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.50e4 + 2.60e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-726. - 1.25e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.45e4 + 5.98e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (6.36e3 - 1.10e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.32e4 + 2.28e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.49e4 - 2.59e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-3.15e4 - 5.47e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.31e5T + 8.58e9T^{2} \) |
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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
−14.27996770606356458563522133591, −12.93634716921204120182622207230, −12.24162417629952061688725924704, −11.96190333731773131015965470997, −9.409587871436116561318335525618, −8.022094653343095520115687458766, −6.12745966490650756690995149637, −5.19340929804151088228086255329, −3.44082392520672163417693293881, −0.65961911552688731003604556412,
3.54925716928676528640512571810, 4.18171557696806928263806147037, 6.03513220868438510420992082140, 7.30367353525533599940121144295, 9.646833326878343745494025034077, 10.92178431947797825648624173650, 11.66841622017078789070909683213, 13.34583360143106514332999467149, 14.17587623833993062264377422521, 15.30845271443475104041004699910
