L(s) = 1 | + (0.133 − 0.5i)3-s + (0.133 − 2.23i)5-s + (2.5 + 0.866i)7-s + (2.36 + 1.36i)9-s + (−1.36 − 2.36i)11-s + (−2 − 2i)13-s + (−1.09 − 0.366i)15-s + (3.73 + i)17-s + (−0.366 + 0.633i)19-s + (0.767 − 1.13i)21-s + (−0.0358 − 0.133i)23-s + (−4.96 − 0.598i)25-s + (2.09 − 2.09i)27-s − 3i·29-s + (6.46 − 3.73i)31-s + ⋯ |
L(s) = 1 | + (0.0773 − 0.288i)3-s + (0.0599 − 0.998i)5-s + (0.944 + 0.327i)7-s + (0.788 + 0.455i)9-s + (−0.411 − 0.713i)11-s + (−0.554 − 0.554i)13-s + (−0.283 − 0.0945i)15-s + (0.905 + 0.242i)17-s + (−0.0839 + 0.145i)19-s + (0.167 − 0.247i)21-s + (−0.00748 − 0.0279i)23-s + (−0.992 − 0.119i)25-s + (0.403 − 0.403i)27-s − 0.557i·29-s + (1.16 − 0.670i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46950 - 0.816006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46950 - 0.816006i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.133 + 2.23i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.73 - i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.366 - 0.633i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.73 - 1.26i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (2.83 - 2.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.83 - 1.83i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.86 - 10.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 + (3.46 - 12.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.83 + 1.63i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.330 - 0.571i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.92 - 5.92i)T - 97iT^{2} \) |
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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
−10.48037414935866491008926133464, −9.857774485482531784108617952347, −8.524426299720178714808624405741, −8.140855207872584245379539498002, −7.27375612590516772429018853883, −5.73541475843463463203343388238, −5.12271736934294828556066785657, −4.08737456179851799988518545782, −2.38552181202392545504218212042, −1.10255016746565741358781149159,
1.70258881407282085513826207091, 3.09642107649895656094075037443, 4.31324408447124397099924329877, 5.14043367461765199068711254715, 6.62309151778698760282845053007, 7.26738058745607335418650563103, 8.065530739955170105230045598795, 9.403688850494770143448678140730, 10.12889241264250841260349528626, 10.70459029094506903220613024829