L(s) = 1 | + (−0.489 + 1.24i)2-s + (−0.370 − 1.69i)3-s + (0.152 + 0.141i)4-s + (−1.59 + 0.770i)5-s + (2.28 + 0.366i)6-s + (1.91 + 1.82i)7-s + (−2.66 + 1.28i)8-s + (−2.72 + 1.25i)9-s + (−0.177 − 2.36i)10-s + (−1.93 − 2.43i)11-s + (0.182 − 0.310i)12-s + (−0.697 + 1.77i)13-s + (−3.21 + 1.48i)14-s + (1.89 + 2.42i)15-s + (−0.264 − 3.53i)16-s + (1.30 − 1.21i)17-s + ⋯ |
L(s) = 1 | + (−0.345 + 0.881i)2-s + (−0.213 − 0.976i)3-s + (0.0761 + 0.0706i)4-s + (−0.715 + 0.344i)5-s + (0.934 + 0.149i)6-s + (0.722 + 0.691i)7-s + (−0.941 + 0.453i)8-s + (−0.908 + 0.417i)9-s + (−0.0561 − 0.749i)10-s + (−0.584 − 0.732i)11-s + (0.0527 − 0.0895i)12-s + (−0.193 + 0.492i)13-s + (−0.859 + 0.397i)14-s + (0.489 + 0.625i)15-s + (−0.0661 − 0.882i)16-s + (0.317 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0922451 + 0.571129i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0922451 + 0.571129i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.370 + 1.69i)T \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
good | 2 | \( 1 + (0.489 - 1.24i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (1.59 - 0.770i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (1.93 + 2.43i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (0.697 - 1.77i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 1.21i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (2.00 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.48 - 6.51i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (8.28 + 2.55i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (1.17 - 2.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.32 + 1.02i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-9.54 - 6.51i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (5.37 - 3.66i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (-2.35 + 6.00i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-5.96 + 1.83i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (4.76 - 3.24i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-5.52 + 5.12i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.50 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.35 - 10.3i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.62 - 0.395i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.61 + 7.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.832 - 2.12i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.06 - 12.9i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-8.43 + 14.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.54775584845508720269541823920, −11.06977415399004081349699455882, −9.334288699080494223313696309288, −8.286273516218381592013948292428, −7.80661814774488720077936167345, −7.17091818216988836878760382015, −5.98506843696288917948957375636, −5.39287540551944005272664253362, −3.41929165617055022031564258492, −2.07114523412737426037328256196,
0.39134815136177240431318426777, 2.33548011984378882971589170813, 3.77631024391008764350475367202, 4.59453769444603812024473388766, 5.69116369570695226418503905947, 7.16015778149191304988665430831, 8.239544410878488515013274260920, 9.144004165737807570744421770172, 10.20461277434596934981340204456, 10.64262019650426380101754202758