L(s) = 1 | + (−0.521 + 1.94i)2-s + 1.44i·3-s + (−1.78 − 1.03i)4-s + (−0.849 − 3.16i)5-s + (−2.81 − 0.753i)6-s + (0.0872 − 0.0872i)8-s + 0.915·9-s + 6.61·10-s + (4.20 − 4.20i)11-s + (1.48 − 2.57i)12-s + (2.81 + 2.25i)13-s + (4.57 − 1.22i)15-s + (−1.93 − 3.35i)16-s + (0.314 − 0.544i)17-s + (−0.477 + 1.78i)18-s + (−0.521 + 0.521i)19-s + ⋯ |
L(s) = 1 | + (−0.368 + 1.37i)2-s + 0.833i·3-s + (−0.892 − 0.515i)4-s + (−0.379 − 1.41i)5-s + (−1.14 − 0.307i)6-s + (0.0308 − 0.0308i)8-s + 0.305·9-s + 2.09·10-s + (1.26 − 1.26i)11-s + (0.429 − 0.743i)12-s + (0.779 + 0.626i)13-s + (1.18 − 0.316i)15-s + (−0.484 − 0.838i)16-s + (0.0762 − 0.132i)17-s + (−0.112 + 0.420i)18-s + (−0.119 + 0.119i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0642 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0642 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.923057 + 0.865530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.923057 + 0.865530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.81 - 2.25i)T \) |
good | 2 | \( 1 + (0.521 - 1.94i)T + (-1.73 - i)T^{2} \) |
| 3 | \( 1 - 1.44iT - 3T^{2} \) |
| 5 | \( 1 + (0.849 + 3.16i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-4.20 + 4.20i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.314 + 0.544i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.521 - 0.521i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.93 + 2.27i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.33 - 2.31i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.06 - 0.285i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (2.44 + 0.655i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.746 + 2.78i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.49 + 2.01i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.10 + 1.90i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.89 - 6.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.919 - 0.246i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 1.08iT - 61T^{2} \) |
| 67 | \( 1 + (3.81 + 3.81i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.590 + 2.20i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.59 + 5.94i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (6.08 - 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.59 + 3.59i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.05 - 3.94i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-8.41 - 2.25i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.66704666195236596400125254017, −9.203065892125411487758839173013, −9.004874666939478456093706077516, −8.435455989476137466166709056878, −7.26833793377801957201065437805, −6.30101118048614919553385019231, −5.40885760298838648277069944489, −4.49130382351582480611079927131, −3.67124154334806223913962439978, −1.02511237451973929791483396223,
1.22486517351914689764618465408, 2.27253349708970913243378670111, 3.36900236340059017372675947526, 4.21919451601443896884220524821, 6.23508995121405682642370475016, 6.91905357598937059095425993419, 7.61149439719623439062220959160, 8.877593521682879476937482291005, 9.857308976676216571902484834403, 10.44550490498187159356508448229