| L(s) = 1 | + (−1.16 + 1.45i)2-s + (0.00644 + 0.0427i)3-s + (−0.329 − 1.44i)4-s + (−0.740 − 1.08i)5-s + (−0.0698 − 0.0403i)6-s + (−2.40 + 1.39i)7-s + (−0.869 − 0.418i)8-s + (2.86 − 0.883i)9-s + (2.44 + 0.183i)10-s + (3.09 + 0.705i)11-s + (0.0596 − 0.0234i)12-s + (−0.0558 − 0.744i)13-s + (0.773 − 5.13i)14-s + (0.0416 − 0.0386i)15-s + (4.29 − 2.06i)16-s + (4.07 + 0.658i)17-s + ⋯ |
| L(s) = 1 | + (−0.822 + 1.03i)2-s + (0.00371 + 0.0246i)3-s + (−0.164 − 0.722i)4-s + (−0.331 − 0.485i)5-s + (−0.0285 − 0.0164i)6-s + (−0.910 + 0.525i)7-s + (−0.307 − 0.148i)8-s + (0.954 − 0.294i)9-s + (0.773 + 0.0579i)10-s + (0.932 + 0.212i)11-s + (0.0172 − 0.00675i)12-s + (−0.0154 − 0.206i)13-s + (0.206 − 1.37i)14-s + (0.0107 − 0.00998i)15-s + (1.07 − 0.517i)16-s + (0.987 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.458376 + 0.700521i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458376 + 0.700521i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-4.07 - 0.658i)T \) |
| 43 | \( 1 + (-5.04 - 4.19i)T \) |
| good | 2 | \( 1 + (1.16 - 1.45i)T + (-0.445 - 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.00644 - 0.0427i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (0.740 + 1.08i)T + (-1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (2.40 - 1.39i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.09 - 0.705i)T + (9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (0.0558 + 0.744i)T + (-12.8 + 1.93i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.423i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (5.99 - 6.45i)T + (-1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.105 + 0.700i)T + (-27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (5.44 - 2.13i)T + (22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-6.51 - 3.76i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.24 - 2.58i)T + (9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-1.72 - 7.55i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.538 - 7.18i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (2.64 - 1.27i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-2.31 - 0.909i)T + (44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-12.0 - 3.70i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (8.87 + 9.56i)T + (-5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (1.02 - 0.0768i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-1.27 + 0.737i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.8 - 1.94i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (5.72 - 0.862i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-9.80 - 2.23i)T + (87.3 + 42.0i)T^{2} \) |
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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
−10.06547193002769256502061081043, −9.562188006246488373482054319713, −9.028210090316924169524852569354, −7.905995307611376465712921899577, −7.38770001280446808878524608462, −6.31708591247802301820937931701, −5.78443007615485204254474886950, −4.26345423460360893277505841614, −3.25268019441973969050618288985, −1.16185845401150688579588009216,
0.70072562773543834303398894059, 2.09190956141215211167679885938, 3.40868101463692699984581664593, 4.06919062561090200549366766402, 5.79294456703198346360499732336, 6.83607830365412856823968671080, 7.56447409874413835468133164312, 8.688739425773410739358555588955, 9.649974979809210295548532018444, 10.03086031744521498231516151826
