L(s) = 1 | − 2-s + (−0.433 − 1.67i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.433 + 1.67i)6-s + (1.23 + 2.33i)7-s − 8-s + (−2.62 + 1.45i)9-s + (0.5 − 0.866i)10-s + (−2.00 − 3.47i)11-s + (−0.433 − 1.67i)12-s + (−0.0260 − 0.0451i)13-s + (−1.23 − 2.33i)14-s + (1.66 + 0.462i)15-s + 16-s + (3.62 − 6.27i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.250 − 0.968i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.177 + 0.684i)6-s + (0.468 + 0.883i)7-s − 0.353·8-s + (−0.874 + 0.485i)9-s + (0.158 − 0.273i)10-s + (−0.605 − 1.04i)11-s + (−0.125 − 0.484i)12-s + (−0.00723 − 0.0125i)13-s + (−0.331 − 0.624i)14-s + (0.430 + 0.119i)15-s + 0.250·16-s + (0.879 − 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.449915 - 0.604673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.449915 - 0.604673i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.433 + 1.67i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.23 - 2.33i)T \) |
good | 11 | \( 1 + (2.00 + 3.47i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0260 + 0.0451i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.62 + 6.27i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0260 + 0.0451i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.65 + 4.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.446 - 0.773i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.776T + 31T^{2} \) |
| 37 | \( 1 + (3.30 + 5.72i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.15 + 10.6i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.24 + 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 8.49T + 47T^{2} \) |
| 53 | \( 1 + (3.85 - 6.67i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 6.00T + 61T^{2} \) |
| 67 | \( 1 - 7.03T + 67T^{2} \) |
| 71 | \( 1 - 2.59T + 71T^{2} \) |
| 73 | \( 1 + (0.00136 - 0.00236i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8.10T + 79T^{2} \) |
| 83 | \( 1 + (0.719 - 1.24i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.125 + 0.218i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0494 + 0.0856i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.68811507754073662655331150785, −9.165091847835781862974560893227, −8.567783235843621598334591889574, −7.65600939379928199665105087785, −7.05116790083416221287864378964, −5.86034244271960766084868368036, −5.24011971007987124262694715975, −3.10720554267598179695983736781, −2.25010280625667741002585228036, −0.56323877189305904094591506741,
1.43755622924931846795747316274, 3.30013868776587544154278398001, 4.38190439891030272979316447518, 5.22348725958884526633224568320, 6.41455678304318441584878474169, 7.68912198001363368362827370918, 8.155884844776377568989543266620, 9.343043919446699563383518161603, 10.02208118076076866741662944130, 10.63657251065399885485004502093