| L(s) = 1 | + (0.109 + 1.40i)2-s + (−0.0225 + 1.73i)3-s + (−1.97 + 0.308i)4-s + (−2.28 − 2.60i)5-s + (−2.44 + 0.157i)6-s + (4.18 + 0.551i)7-s + (−0.651 − 2.75i)8-s + (−2.99 − 0.0780i)9-s + (3.41 − 3.50i)10-s + (−1.27 − 2.58i)11-s + (−0.489 − 3.42i)12-s + (−2.06 − 6.06i)13-s + (−0.319 + 5.96i)14-s + (4.55 − 3.89i)15-s + (3.80 − 1.21i)16-s + (−3.99 − 3.99i)17-s + ⋯ |
| L(s) = 1 | + (0.0773 + 0.997i)2-s + (−0.0130 + 0.999i)3-s + (−0.988 + 0.154i)4-s + (−1.02 − 1.16i)5-s + (−0.997 + 0.0644i)6-s + (1.58 + 0.208i)7-s + (−0.230 − 0.973i)8-s + (−0.999 − 0.0260i)9-s + (1.08 − 1.10i)10-s + (−0.384 − 0.779i)11-s + (−0.141 − 0.989i)12-s + (−0.571 − 1.68i)13-s + (−0.0852 + 1.59i)14-s + (1.17 − 1.00i)15-s + (0.952 − 0.304i)16-s + (−0.968 − 0.968i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.784605 - 0.141811i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.784605 - 0.141811i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.109 - 1.40i)T \) |
| 3 | \( 1 + (0.0225 - 1.73i)T \) |
| good | 5 | \( 1 + (2.28 + 2.60i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-4.18 - 0.551i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.27 + 2.58i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (2.06 + 6.06i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (3.99 + 3.99i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.252 + 1.26i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.851 - 6.46i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 0.0698i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (1.47 - 0.850i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.401 - 2.01i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.989 + 7.51i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-0.927 + 0.457i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (3.04 - 0.816i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.80 - 2.69i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (1.47 + 1.67i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.524 + 8.00i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-0.462 - 0.228i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (6.15 + 14.8i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.653 - 1.57i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-3.20 - 11.9i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.225 - 0.197i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-10.0 + 4.17i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (13.1 + 7.61i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.73783493192804497129624964150, −9.426857368296587242047610235044, −8.658258301164587406265591222932, −8.098785109150872637066757000291, −7.51249849258151409475836444320, −5.55430402832443102848942492352, −5.08038474162899791888389998975, −4.51983433717065523514973367513, −3.28553801708910304043984624656, −0.44895632414517018682372148845,
1.75614383640203513182684257080, 2.51972940507169626944379904904, 4.10636510650631657418274323774, 4.76675154173982861952268630787, 6.46173464177578566681899027168, 7.35049548722114330475938288022, 8.084409861022182635619429071398, 8.858278542319848503920677515168, 10.34501823278427422281207152030, 11.07987411683520072322166889540
