L(s) = 1 | − 2.84i·3-s + 5-s + (0.712 + 2.54i)7-s − 5.08·9-s − 5.41·11-s − 0.641·13-s − 2.84i·15-s − 5.16i·17-s − 3.44i·19-s + (7.24 − 2.02i)21-s + 7.94i·23-s + 25-s + 5.94i·27-s − 10.0i·29-s − 5.38·31-s + ⋯ |
L(s) = 1 | − 1.64i·3-s + 0.447·5-s + (0.269 + 0.963i)7-s − 1.69·9-s − 1.63·11-s − 0.177·13-s − 0.734i·15-s − 1.25i·17-s − 0.790i·19-s + (1.58 − 0.442i)21-s + 1.65i·23-s + 0.200·25-s + 1.14i·27-s − 1.86i·29-s − 0.966·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2161787008\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2161787008\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-0.712 - 2.54i)T \) |
good | 3 | \( 1 + 2.84iT - 3T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 0.641T + 13T^{2} \) |
| 17 | \( 1 + 5.16iT - 17T^{2} \) |
| 19 | \( 1 + 3.44iT - 19T^{2} \) |
| 23 | \( 1 - 7.94iT - 23T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 - 0.931iT - 37T^{2} \) |
| 41 | \( 1 - 8.33iT - 41T^{2} \) |
| 43 | \( 1 + 6.76T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 - 6.31iT - 53T^{2} \) |
| 59 | \( 1 - 11.5iT - 59T^{2} \) |
| 61 | \( 1 + 0.834T + 61T^{2} \) |
| 67 | \( 1 + 2.01T + 67T^{2} \) |
| 71 | \( 1 - 0.628iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 2.98iT - 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 13.3iT - 89T^{2} \) |
| 97 | \( 1 + 9.07iT - 97T^{2} \) |
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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
−8.204570990515840802552866504063, −7.74594009702316911913949477750, −7.11280704045685085546042519963, −6.16413322020621297190321406560, −5.49589614749653809813913695163, −4.89773820627250146736945229929, −2.91894766038879967981977308551, −2.48397339599397373190993179994, −1.54214507374815825634115791590, −0.06790488472453940690836189446,
1.93192254246177291498633874441, 3.22870864493752491711024784078, 3.86751723905767243083211739186, 4.90764402818295282338246631979, 5.20180804286587726592165005895, 6.25384495472044974748605533848, 7.30768835115873373007581015564, 8.312182750246184771605995308375, 8.717453393979194483299501038402, 9.934845101161714025163738285923