L(s) = 1 | + (−1.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (1.5 − 2.59i)9-s + (2 + 3.46i)13-s + (1.5 + 0.866i)15-s − 6·17-s − 2·19-s − 1.73i·21-s + (−1.5 − 2.59i)23-s + (−0.499 + 0.866i)25-s + 5.19i·27-s + (−1.5 + 2.59i)29-s + (−5 − 8.66i)31-s + 0.999·35-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.188 + 0.327i)7-s + (0.5 − 0.866i)9-s + (0.554 + 0.960i)13-s + (0.387 + 0.223i)15-s − 1.45·17-s − 0.458·19-s − 0.377i·21-s + (−0.312 − 0.541i)23-s + (−0.0999 + 0.173i)25-s + 0.999i·27-s + (−0.278 + 0.482i)29-s + (−0.898 − 1.55i)31-s + 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.12004617562558317612234555629, −9.049173408442582028060569306632, −8.693431316048271361538352350262, −7.15346157054947000600534800765, −6.41203781582680632548827353130, −5.51434012765184138238758748848, −4.47202894275194399058100555398, −3.77857782832053877074108713568, −1.94778818271643003958566757316, 0,
1.73870141948584193105546101954, 3.28228619378010980214582817528, 4.49613787151169548431890054640, 5.55427569378275538933118585294, 6.47477624670373675122207936470, 7.13070494244411027727927907447, 8.050711853716119625023850986512, 9.012758654883809749389358977690, 10.41940890496104677867537904419