| L(s) = 1 | + (−0.104 − 0.994i)2-s + (−1.72 + 0.119i)3-s + (−0.978 + 0.207i)4-s + (−1.30 − 2.92i)5-s + (0.299 + 1.70i)6-s + (−2.04 − 1.67i)7-s + (0.309 + 0.951i)8-s + (2.97 − 0.412i)9-s + (−2.77 + 1.60i)10-s + (0.980 − 3.16i)11-s + (1.66 − 0.476i)12-s + (−0.203 − 0.280i)13-s + (−1.45 + 2.20i)14-s + (2.59 + 4.89i)15-s + (0.913 − 0.406i)16-s + (−0.764 + 7.27i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (−0.997 + 0.0689i)3-s + (−0.489 + 0.103i)4-s + (−0.582 − 1.30i)5-s + (0.122 + 0.696i)6-s + (−0.773 − 0.634i)7-s + (0.109 + 0.336i)8-s + (0.990 − 0.137i)9-s + (−0.876 + 0.506i)10-s + (0.295 − 0.955i)11-s + (0.480 − 0.137i)12-s + (−0.0565 − 0.0778i)13-s + (−0.388 + 0.590i)14-s + (0.671 + 1.26i)15-s + (0.228 − 0.101i)16-s + (−0.185 + 1.76i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0895839 + 0.116755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0895839 + 0.116755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 3 | \( 1 + (1.72 - 0.119i)T \) |
| 7 | \( 1 + (2.04 + 1.67i)T \) |
| 11 | \( 1 + (-0.980 + 3.16i)T \) |
| good | 5 | \( 1 + (1.30 + 2.92i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (0.203 + 0.280i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.764 - 7.27i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (1.45 - 6.85i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (6.50 + 3.75i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.673 + 2.07i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.996 + 0.443i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.821 - 0.912i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.62 - 4.98i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.84iT - 43T^{2} \) |
| 47 | \( 1 + (-0.790 + 3.71i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.42 + 5.44i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.23 + 5.80i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.0447 - 0.100i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-5.60 - 9.70i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.42 - 4.71i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.794 + 3.73i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (6.66 - 0.700i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (9.88 + 7.18i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (1.06 + 0.617i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.00 - 5.81i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.40008359728042104123599428678, −9.891741718870520087279450970912, −8.575600170078223128503339366979, −7.992083300764208289452108721747, −6.37477638439797875835358343683, −5.65106066683772319482253861861, −4.15284791512679207341198883882, −3.89247794986480079274438385960, −1.41324654591015179955475510295, −0.11168870976042020768779574427,
2.64836601169160362656095657536, 4.13359946298869128048614769281, 5.24668040476237343501975125648, 6.36701376209665235213647778106, 7.02496767061396358017790478169, 7.48418289239264589712276095139, 9.192017621943357093113856854850, 9.826804738224156889677632355643, 10.85920918892560485255654383864, 11.67519734629417600560019218809
