L(s) = 1 | + (−1.18 + 1.63i)2-s + (2.77 + 0.900i)3-s + (−0.646 − 1.98i)4-s + (0.135 + 2.23i)5-s + (−4.76 + 3.46i)6-s + (3.05 − 0.992i)7-s + (0.176 + 0.0573i)8-s + (4.44 + 3.23i)9-s + (−3.81 − 2.43i)10-s − 6.09i·12-s + (−0.381 + 0.524i)13-s + (−2.00 + 6.17i)14-s + (−1.63 + 6.30i)15-s + (3.08 − 2.23i)16-s + (0.698 + 0.961i)17-s + (−10.5 + 3.43i)18-s + ⋯ |
L(s) = 1 | + (−0.840 + 1.15i)2-s + (1.60 + 0.520i)3-s + (−0.323 − 0.994i)4-s + (0.0607 + 0.998i)5-s + (−1.94 + 1.41i)6-s + (1.15 − 0.375i)7-s + (0.0624 + 0.0202i)8-s + (1.48 + 1.07i)9-s + (−1.20 − 0.768i)10-s − 1.76i·12-s + (−0.105 + 0.145i)13-s + (−0.536 + 1.65i)14-s + (−0.421 + 1.62i)15-s + (0.770 − 0.559i)16-s + (0.169 + 0.233i)17-s + (−2.49 + 0.809i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.574748 + 1.68019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.574748 + 1.68019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.135 - 2.23i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.18 - 1.63i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-2.77 - 0.900i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-3.05 + 0.992i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (0.381 - 0.524i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.698 - 0.961i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.584 - 1.79i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.35iT - 23T^{2} \) |
| 29 | \( 1 + (-1.28 - 3.96i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.39 + 4.64i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.95 + 0.636i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.02 - 3.14i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (-2.77 - 0.900i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.77 + 5.19i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.14 - 3.53i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 4.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 2.37iT - 67T^{2} \) |
| 71 | \( 1 + (12.7 - 9.23i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.43 - 1.76i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (12.3 + 8.96i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (6.25 + 8.60i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-8.89 + 12.2i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.42814097167475685630625151049, −9.963043542342005373301358549155, −8.854893321699758970757232683068, −8.406968646794718713033651185100, −7.52361340235985442567814417389, −7.12464381247814834263143448535, −5.78411416276428091195481018920, −4.35455125571913382750100064384, −3.29479532207690743240600380791, −2.02221323506045293003543024860,
1.25611839201433283776978607852, 2.00141577914087174975622285702, 3.01345924102800128399008394941, 4.25890037597681258055146124403, 5.56517469762431572969445314539, 7.40495005930287711357497968919, 8.135855959710424777089540233593, 8.705018064870302786352253875430, 9.255736171014432145935027543313, 10.00524131814761676045341302541