| L(s) = 1 | + (−0.999 + 0.726i)2-s + (1.53 − 0.844i)3-s + (−0.146 + 0.450i)4-s + (−2.06 + 3.25i)5-s + (−0.922 + 1.96i)6-s + (−2.24 + 0.886i)7-s + (−0.944 − 2.90i)8-s + (0.0408 − 0.0644i)9-s + (−0.299 − 4.75i)10-s + (4.10 + 2.25i)11-s + (0.155 + 0.815i)12-s + (0.0152 + 0.0142i)13-s + (1.59 − 2.51i)14-s + (−0.425 + 6.75i)15-s + (2.28 + 1.66i)16-s + (1.97 + 0.783i)17-s + ⋯ |
| L(s) = 1 | + (−0.706 + 0.513i)2-s + (0.887 − 0.487i)3-s + (−0.0731 + 0.225i)4-s + (−0.924 + 1.45i)5-s + (−0.376 + 0.800i)6-s + (−0.846 + 0.335i)7-s + (−0.333 − 1.02i)8-s + (0.0136 − 0.0214i)9-s + (−0.0946 − 1.50i)10-s + (1.23 + 0.679i)11-s + (0.0449 + 0.235i)12-s + (0.00422 + 0.00396i)13-s + (0.426 − 0.671i)14-s + (−0.109 + 1.74i)15-s + (0.572 + 0.415i)16-s + (0.480 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.418465 + 0.640923i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.418465 + 0.640923i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (1.82 + 12.1i)T \) |
| good | 2 | \( 1 + (0.999 - 0.726i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.53 + 0.844i)T + (1.60 - 2.53i)T^{2} \) |
| 5 | \( 1 + (2.06 - 3.25i)T + (-2.12 - 4.52i)T^{2} \) |
| 7 | \( 1 + (2.24 - 0.886i)T + (5.10 - 4.79i)T^{2} \) |
| 11 | \( 1 + (-4.10 - 2.25i)T + (5.89 + 9.28i)T^{2} \) |
| 13 | \( 1 + (-0.0152 - 0.0142i)T + (0.816 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.97 - 0.783i)T + (12.3 + 11.6i)T^{2} \) |
| 19 | \( 1 + (-2.36 + 7.26i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.86 - 8.82i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.72 + 0.343i)T + (28.0 - 7.21i)T^{2} \) |
| 31 | \( 1 + (-8.77 + 2.25i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (4.67 + 4.38i)T + (2.32 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-0.692 + 1.47i)T + (-26.1 - 31.5i)T^{2} \) |
| 43 | \( 1 + (-3.62 - 1.43i)T + (31.3 + 29.4i)T^{2} \) |
| 47 | \( 1 + (-0.363 + 0.771i)T + (-29.9 - 36.2i)T^{2} \) |
| 53 | \( 1 + (-0.612 + 3.21i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (6.39 + 4.64i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.48 - 1.91i)T + (32.6 + 51.5i)T^{2} \) |
| 67 | \( 1 + (-1.13 - 1.79i)T + (-28.5 + 60.6i)T^{2} \) |
| 71 | \( 1 + (-4.24 + 1.68i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (-2.76 + 1.09i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (1.62 + 0.416i)T + (69.2 + 38.0i)T^{2} \) |
| 83 | \( 1 + (9.16 - 11.0i)T + (-15.5 - 81.5i)T^{2} \) |
| 89 | \( 1 + (-6.12 + 7.40i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-3.85 - 6.07i)T + (-41.3 + 87.7i)T^{2} \) |
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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
−13.51219572394881807024494652408, −12.25812724389751694304454472367, −11.41474414893697516336326780595, −9.861485761918930178997038695201, −9.085987254332095952187370041781, −7.88570620013459656876818479201, −7.20205978216148416826365377272, −6.51758783083256368572925131497, −3.75721689769318494754113971995, −2.87164300970481210887863035259,
0.923340110519782439112529964258, 3.36035142823725034603516182551, 4.43989106868389392986924289682, 6.13359173133860541249830346550, 8.222802027611136978340405608782, 8.601983409520891322406379178546, 9.528421489720253535321102874320, 10.25413461099896366955615895536, 11.87715557872948003461963668188, 12.27512876749794188633105428990
