L(s) = 1 | + (0.765 − 1.32i)2-s + (−0.5 − 0.866i)3-s + (−0.171 − 0.297i)4-s + (0.5 − 0.866i)5-s − 1.53·6-s + (1.95 − 1.78i)7-s + 2.53·8-s + (−0.499 + 0.866i)9-s + (−0.765 − 1.32i)10-s + (0.5 + 0.866i)11-s + (−0.171 + 0.297i)12-s + 5.35·13-s + (−0.865 − 3.95i)14-s − 0.999·15-s + (2.28 − 3.95i)16-s + (0.656 + 1.13i)17-s + ⋯ |
L(s) = 1 | + (0.541 − 0.937i)2-s + (−0.288 − 0.499i)3-s + (−0.0857 − 0.148i)4-s + (0.223 − 0.387i)5-s − 0.624·6-s + (0.739 − 0.673i)7-s + 0.896·8-s + (−0.166 + 0.288i)9-s + (−0.242 − 0.419i)10-s + (0.150 + 0.261i)11-s + (−0.0495 + 0.0857i)12-s + 1.48·13-s + (−0.231 − 1.05i)14-s − 0.258·15-s + (0.571 − 0.989i)16-s + (0.159 + 0.275i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.635856372\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.635856372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.95 + 1.78i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.765 + 1.32i)T + (-1 - 1.73i)T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 + (-0.656 - 1.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.94 - 5.10i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.460 + 0.798i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + (1.75 + 3.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.60 - 4.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + (0.978 - 1.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.06 + 8.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.32 + 9.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.0865 + 0.149i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.55 - 4.42i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 + (-0.192 - 0.333i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.05 - 8.75i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + (-0.866 + 1.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 0.868T + 97T^{2} \) |
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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
−10.02093960829520766716834899561, −8.388298618669615525273831946824, −8.191552572863778041877421499776, −6.98862875500344968189024963232, −6.14073068509793408741560572423, −5.01600968659821964217914800631, −4.20472431647529938540917316035, −3.35985909111341877546741233952, −1.85017917305758631402827343505, −1.25444130990879436141552405252,
1.51270400880296205069306149967, 3.03327164766817991473704523007, 4.30070915886927000266538349546, 5.04507022933425624041215488748, 5.86772460733354706150048941771, 6.44097811227577473996534908157, 7.29900271210130226950274053730, 8.493800575143539419367089732443, 8.884648036831368715388070775552, 10.20626440110160417049456978752