| L(s) = 1 | + (−2.47 − 0.436i)2-s + (−1.50 + 0.853i)3-s + (4.05 + 1.47i)4-s + (−2.58 − 0.939i)5-s + (4.10 − 1.45i)6-s + (−2.62 + 0.338i)7-s + (−5.05 − 2.91i)8-s + (1.54 − 2.57i)9-s + (5.97 + 3.45i)10-s + (1.43 + 3.93i)11-s + (−7.37 + 1.23i)12-s + (0.0398 − 0.109i)13-s + (6.64 + 0.307i)14-s + (4.69 − 0.786i)15-s + (4.61 + 3.87i)16-s + (2.57 − 4.45i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.308i)2-s + (−0.870 + 0.492i)3-s + (2.02 + 0.738i)4-s + (−1.15 − 0.420i)5-s + (1.67 − 0.593i)6-s + (−0.991 + 0.127i)7-s + (−1.78 − 1.03i)8-s + (0.514 − 0.857i)9-s + (1.89 + 1.09i)10-s + (0.431 + 1.18i)11-s + (−2.13 + 0.357i)12-s + (0.0110 − 0.0303i)13-s + (1.77 + 0.0823i)14-s + (1.21 − 0.202i)15-s + (1.15 + 0.967i)16-s + (0.624 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.267715 - 0.0586319i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.267715 - 0.0586319i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 - 0.853i)T \) |
| 7 | \( 1 + (2.62 - 0.338i)T \) |
| good | 2 | \( 1 + (2.47 + 0.436i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (2.58 + 0.939i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.43 - 3.93i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.0398 + 0.109i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.57 + 4.45i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.51 + 2.60i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.39 + 1.30i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.21 - 3.34i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.106 - 0.291i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + (6.29 + 2.29i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.411 + 2.33i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (4.74 - 1.72i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.76 + 2.17i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.02 + 6.73i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.395 + 1.08i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.92 - 10.9i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.47 - 3.15i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.58iT - 73T^{2} \) |
| 79 | \( 1 + (-2.74 + 15.5i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-10.6 + 3.88i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (0.456 + 0.791i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.88 - 1.03i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.90088586617318661354437297615, −11.52185056728784761434368465928, −10.29031514042208074739977705172, −9.554503642919373608302062061865, −8.863522464184409659547897966422, −7.33005231181494816905647770004, −6.87596106264125819091529867028, −4.93486913159273402308697761746, −3.27285392755926884183465812644, −0.70358244678819500507517425061,
0.914899154936491779272640670624, 3.40524695475566898186328685177, 5.87484945808189344052522369849, 6.75182389204953207749991327821, 7.58772893049738856165697673020, 8.412427207747138200308468556592, 9.699616140483019927699487769866, 10.65123773918164170399003024114, 11.37322195501574887943759141901, 12.09938328047315320968210580005
