L(s) = 1 | + 14.6i·2-s + (2.75 − 26.8i)3-s − 149.·4-s + 100. i·5-s + (392. + 40.2i)6-s − 263.·7-s − 1.25e3i·8-s + (−713. − 147. i)9-s − 1.47e3·10-s − 497. i·11-s + (−412. + 4.02e3i)12-s + 2.89e3·13-s − 3.84e3i·14-s + (2.70e3 + 277. i)15-s + 8.79e3·16-s − 8.32e3i·17-s + ⋯ |
L(s) = 1 | + 1.82i·2-s + (0.101 − 0.994i)3-s − 2.34·4-s + 0.806i·5-s + (1.81 + 0.186i)6-s − 0.767·7-s − 2.45i·8-s + (−0.979 − 0.202i)9-s − 1.47·10-s − 0.373i·11-s + (−0.238 + 2.33i)12-s + 1.31·13-s − 1.40i·14-s + (0.802 + 0.0821i)15-s + 2.14·16-s − 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.923893 - 0.0471854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.923893 - 0.0471854i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.75 + 26.8i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
good | 2 | \( 1 - 14.6iT - 64T^{2} \) |
| 5 | \( 1 - 100. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 263.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 497. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.89e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 8.32e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.36e3T + 4.70e7T^{2} \) |
| 29 | \( 1 + 2.83e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.76e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 4.32e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 5.16e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.65e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.17e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.15e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.11e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.70e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.84e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 1.53e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.50e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.57e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 7.94e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.54e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.35e6T + 8.32e11T^{2} \) |
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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
−13.77546642611360054761801103350, −13.06852971302365433022793121449, −11.39559127869922129651946082031, −9.505607696442552259488121033825, −8.391073974393162898825401969001, −7.20456959085165976458674217037, −6.56838431488630702150447807004, −5.54082773942434752187847106429, −3.29596918002072964881340722308, −0.39660828524867063776990960150,
1.38657096441042182120812482953, 3.28294998617259525316328216243, 4.13238288655832623951278903308, 5.54692074648124161119904774715, 8.613881434936337972899818136360, 9.225068254737938925111816938634, 10.34561054966100857434587894650, 11.06179418073849493175608341177, 12.40701180413020439374768718275, 13.07357554624768267321600559563