| L(s) = 1 | + (0.540 + 0.841i)2-s + (−1.71 − 0.210i)3-s + (−0.415 + 0.909i)4-s + (−2.84 + 0.836i)5-s + (−0.752 − 1.56i)6-s + (−3.02 + 2.61i)7-s + (−0.989 + 0.142i)8-s + (2.91 + 0.724i)9-s + (−2.24 − 1.94i)10-s + (2.79 + 1.79i)11-s + (0.905 − 1.47i)12-s + (1.31 − 1.51i)13-s + (−3.83 − 1.12i)14-s + (5.07 − 0.838i)15-s + (−0.654 − 0.755i)16-s + (0.139 + 0.306i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (−0.992 − 0.121i)3-s + (−0.207 + 0.454i)4-s + (−1.27 + 0.374i)5-s + (−0.307 − 0.636i)6-s + (−1.14 + 0.990i)7-s + (−0.349 + 0.0503i)8-s + (0.970 + 0.241i)9-s + (−0.709 − 0.614i)10-s + (0.842 + 0.541i)11-s + (0.261 − 0.426i)12-s + (0.365 − 0.421i)13-s + (−1.02 − 0.301i)14-s + (1.31 − 0.216i)15-s + (−0.163 − 0.188i)16-s + (0.0339 + 0.0742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112077 + 0.522950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112077 + 0.522950i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 + (1.71 + 0.210i)T \) |
| 23 | \( 1 + (0.384 - 4.78i)T \) |
| good | 5 | \( 1 + (2.84 - 0.836i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (3.02 - 2.61i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.79 - 1.79i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.31 + 1.51i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.139 - 0.306i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (4.42 + 2.02i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-6.14 + 2.80i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.10 - 7.71i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.31 - 7.87i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.321 - 1.09i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.439 + 0.0632i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 8.33iT - 47T^{2} \) |
| 53 | \( 1 + (0.581 + 0.671i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (3.38 + 2.93i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-7.37 + 1.05i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (6.97 + 10.8i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (6.79 + 10.5i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.56 + 9.98i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-11.8 - 10.2i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.91 - 1.73i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.63 - 11.3i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 6.41i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−13.44240024834117346665363191513, −12.25622527600190607786357396922, −12.06780360550809951858149233636, −10.81883602120209382031884685262, −9.443495927970520849807844996911, −8.103644386957523022648629786113, −6.80352307785495593442655198986, −6.23414724284317760007068042581, −4.72773700123886630800957823770, −3.40477969806025601441635977849,
0.55216684260917041913160641516, 3.80490395720517363370855001473, 4.28948328987083370255711838071, 6.12838644002887083995447831626, 7.04057606592916580210167132768, 8.685235379599028580482625775743, 10.05713339508238525973686857191, 10.89528389067784415279135993868, 11.80710118998357961135039044509, 12.50801448288297777791189549003
