L(s) = 1 | + (−0.129 + 0.0745i)2-s + (3.24 + 5.61i)3-s + (−3.98 + 6.90i)4-s + 10.2i·5-s + (−0.837 − 0.483i)6-s + (25.6 + 14.8i)7-s − 2.38i·8-s + (−7.55 + 13.0i)9-s + (−0.764 − 1.32i)10-s + (32.9 − 19.0i)11-s − 51.7·12-s − 4.42·14-s + (−57.6 + 33.2i)15-s + (−31.7 − 54.9i)16-s + (−35.6 + 61.7i)17-s − 2.25i·18-s + ⋯ |
L(s) = 1 | + (−0.0456 + 0.0263i)2-s + (0.624 + 1.08i)3-s + (−0.498 + 0.863i)4-s + 0.917i·5-s + (−0.0569 − 0.0329i)6-s + (1.38 + 0.801i)7-s − 0.105i·8-s + (−0.279 + 0.484i)9-s + (−0.0241 − 0.0418i)10-s + (0.904 − 0.522i)11-s − 1.24·12-s − 0.0844·14-s + (−0.991 + 0.572i)15-s + (−0.495 − 0.858i)16-s + (−0.508 + 0.880i)17-s − 0.0294i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.732233 + 1.96585i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.732233 + 1.96585i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (0.129 - 0.0745i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (-3.24 - 5.61i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 10.2iT - 125T^{2} \) |
| 7 | \( 1 + (-25.6 - 14.8i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-32.9 + 19.0i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (35.6 - 61.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-8.74 - 5.04i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (99.3 + 172. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (15.4 + 26.6i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 151. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-131. + 75.6i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-179. + 103. i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (151. - 262. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 12.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (338. + 195. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-78.2 + 135. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-263. + 151. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-791. - 456. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + 249. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.02e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (819. - 473. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-361. - 208. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.61881154291462370615063698534, −11.54217835582285221881392976597, −10.75281319380800788671317902842, −9.474516483353598058313136051879, −8.607027973611158171612734246680, −7.992534130041682201391407232707, −6.33908299709976001088073610490, −4.62447299064145182443891764700, −3.80061060701876548432138402179, −2.49401839899275658640763122227,
1.07740028819914342822537702971, 1.73169256558626728322960186368, 4.31554061166437277441795375856, 5.20128529922335189377150849297, 6.86576776287021268852537204725, 7.85235640273387391271302555237, 8.762844045657744625347616597403, 9.676116318582253397455254591493, 11.05419996773245439752938032603, 12.03378104909499523083914156836