| L(s) = 1 | + (1.35 + 0.420i)2-s + (1.30 + 0.456i)3-s + (1.64 + 1.13i)4-s + (−3.64 − 0.831i)5-s + (1.56 + 1.16i)6-s + (−0.00558 − 0.0115i)7-s + (1.74 + 2.22i)8-s + (−0.854 − 0.681i)9-s + (−4.56 − 2.65i)10-s + (0.482 − 4.28i)11-s + (1.62 + 2.23i)12-s + (−1.91 + 1.52i)13-s + (−0.00266 − 0.0180i)14-s + (−4.36 − 2.74i)15-s + (1.41 + 3.73i)16-s + (0.542 − 0.542i)17-s + ⋯ |
| L(s) = 1 | + (0.954 + 0.297i)2-s + (0.752 + 0.263i)3-s + (0.823 + 0.567i)4-s + (−1.62 − 0.371i)5-s + (0.640 + 0.475i)6-s + (−0.00211 − 0.00438i)7-s + (0.616 + 0.786i)8-s + (−0.284 − 0.227i)9-s + (−1.44 − 0.839i)10-s + (0.145 − 1.29i)11-s + (0.469 + 0.644i)12-s + (−0.530 + 0.423i)13-s + (−0.000711 − 0.00481i)14-s + (−1.12 − 0.708i)15-s + (0.354 + 0.934i)16-s + (0.131 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65400 + 0.429919i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65400 + 0.429919i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.35 - 0.420i)T \) |
| 29 | \( 1 + (5.34 - 0.662i)T \) |
| good | 3 | \( 1 + (-1.30 - 0.456i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (3.64 + 0.831i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (0.00558 + 0.0115i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.482 + 4.28i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (1.91 - 1.52i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.542 + 0.542i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.93 - 5.53i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (-5.63 + 1.28i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (6.05 - 3.80i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (3.39 - 0.382i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-0.383 - 0.383i)T + 41iT^{2} \) |
| 43 | \( 1 + (-0.0943 + 0.150i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-8.20 - 0.924i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (-0.266 + 1.16i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 - 6.21iT - 59T^{2} \) |
| 61 | \( 1 + (-2.56 + 7.32i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (-5.52 + 6.92i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (3.36 + 4.21i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (7.79 - 12.4i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 1.19i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (1.85 - 3.85i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-3.61 - 5.74i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (2.17 + 6.22i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−13.89791959324177769228690441218, −12.56772202012835562256868870255, −11.78569679203502735222936889782, −10.96499392577088337184425881209, −8.949093758350673167890896461578, −8.138381794983148143902503660410, −7.14963611643848367406531416497, −5.46154597117757937254193569384, −3.93974950043858905836209509994, −3.25510285875019851204974213085,
2.61021918839809482564447466468, 3.81392731468888399464852972952, 5.08386472700561588534100381867, 7.26703041884214372622359781834, 7.48028102358916946376684387451, 9.182137129015482798652794498238, 10.76360023341567027599001635671, 11.57279089163468362160187644615, 12.50905670091871773328153871704, 13.38365414341794294021130449417
