| L(s) = 1 | + (0.854 − 0.620i)2-s + (7.01 + 21.5i)3-s + (−9.54 + 29.3i)4-s + (20.2 + 14.6i)5-s + (19.3 + 14.0i)6-s + (3.78 − 11.6i)7-s + (20.5 + 63.1i)8-s + (−219. + 159. i)9-s + 26.3·10-s + (−309. − 255. i)11-s − 700.·12-s + (149. − 108. i)13-s + (−3.99 − 12.2i)14-s + (−175. + 539. i)15-s + (−742. − 539. i)16-s + (558. + 405. i)17-s + ⋯ |
| L(s) = 1 | + (0.151 − 0.109i)2-s + (0.449 + 1.38i)3-s + (−0.298 + 0.917i)4-s + (0.361 + 0.262i)5-s + (0.219 + 0.159i)6-s + (0.0291 − 0.0898i)7-s + (0.113 + 0.348i)8-s + (−0.904 + 0.657i)9-s + 0.0834·10-s + (−0.771 − 0.635i)11-s − 1.40·12-s + (0.244 − 0.177i)13-s + (−0.00544 − 0.0167i)14-s + (−0.201 + 0.618i)15-s + (−0.725 − 0.527i)16-s + (0.468 + 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.647635 + 1.73943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647635 + 1.73943i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-20.2 - 14.6i)T \) |
| 11 | \( 1 + (309. + 255. i)T \) |
| good | 2 | \( 1 + (-0.854 + 0.620i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (-7.01 - 21.5i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + (-3.78 + 11.6i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-149. + 108. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-558. - 405. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-603. - 1.85e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 23.9T + 6.43e6T^{2} \) |
| 29 | \( 1 + (1.40e3 - 4.32e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-3.16e3 + 2.29e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.44e3 + 7.53e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (94.1 + 289. i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 7.17e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.80e3 - 5.55e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.10e4 - 8.06e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.19e4 + 3.68e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.49e4 - 3.26e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + 5.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.87e4 + 1.36e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.03e4 - 6.25e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-3.43e4 + 2.49e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.96e4 - 5.05e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 5.94e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.13e4 - 3.72e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.66819259581972309539100301621, −13.78492364129133937904911146446, −12.57248299450634347723063722271, −11.03462246244767186499709858119, −10.06059194017216300218555936220, −8.877209337414315887157832908273, −7.79008187125693537855208439316, −5.51629212447645244508592090808, −4.00328351653418031937326107313, −2.97531580139448935306047949191,
0.869710536644759031865064369904, 2.30059054635455869282513817810, 4.99447242184227944574897949371, 6.38644509240278019881647058742, 7.57412734895910640459637186007, 8.977186771539180638360655639996, 10.20359217617497218795089875385, 11.84447110313067612684830141091, 13.20125214411974560852791571936, 13.56756304401445378767223311030
