| L(s) = 1 | + (−0.284 + 1.97i)2-s + (3.50 + 7.66i)3-s + (−3.83 − 1.12i)4-s + (−3.27 − 3.77i)5-s + (−16.1 + 4.74i)6-s + (−21.3 − 13.7i)7-s + (3.32 − 7.27i)8-s + (−28.8 + 33.2i)9-s + (8.41 − 5.40i)10-s + (−7.27 − 50.6i)11-s + (−4.79 − 33.3i)12-s + (39.2 − 25.2i)13-s + (33.2 − 38.3i)14-s + (17.5 − 38.3i)15-s + (13.4 + 8.65i)16-s + (8.31 − 2.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.673 + 1.47i)3-s + (−0.479 − 0.140i)4-s + (−0.292 − 0.337i)5-s + (−1.10 + 0.323i)6-s + (−1.15 − 0.741i)7-s + (0.146 − 0.321i)8-s + (−1.06 + 1.23i)9-s + (0.266 − 0.170i)10-s + (−0.199 − 1.38i)11-s + (−0.115 − 0.802i)12-s + (0.837 − 0.538i)13-s + (0.635 − 0.732i)14-s + (0.301 − 0.659i)15-s + (0.210 + 0.135i)16-s + (0.118 − 0.0348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.395462 - 0.221825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.395462 - 0.221825i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 - 1.97i)T \) |
| 5 | \( 1 + (3.27 + 3.77i)T \) |
| 23 | \( 1 + (6.56 - 110. i)T \) |
| good | 3 | \( 1 + (-3.50 - 7.66i)T + (-17.6 + 20.4i)T^{2} \) |
| 7 | \( 1 + (21.3 + 13.7i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (7.27 + 50.6i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (-39.2 + 25.2i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (-8.31 + 2.44i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (131. + 38.5i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (289. - 84.9i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-18.9 + 41.5i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (10.9 - 12.5i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-135. - 156. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (148. + 324. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (466. + 299. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-717. + 461. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (175. - 384. i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-3.92 + 27.3i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-104. + 727. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (193. + 56.8i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-75.4 + 48.4i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-873. + 1.00e3i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (-224. - 492. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (452. + 522. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−11.12130604015007800186281200148, −10.47238475414085971661874272164, −9.449127231197054977396801274774, −8.766455383382679507611487392677, −7.902692801303581584804240650297, −6.39341383496795612885936173822, −5.28159332067771470704047436730, −3.85152217816426753306211322753, −3.40528494359270752683091675344, −0.16860222062841065595665113563,
1.82867080476322189099339070084, 2.65840528867063750014451101757, 4.01119858886932159241327006541, 6.14197940908358334206269789810, 6.91407468311841629692557486767, 8.036420457022002385842923659333, 8.928096477946248268307123761975, 9.868547467261704058745054884368, 11.14396562392121958405621664273, 12.37912213096079751739503491122
