| L(s) = 1 | + (−0.569 − 1.33i)2-s + (1.33 − 0.243i)3-s + (−0.0679 + 0.0710i)4-s + (0.862 + 2.65i)5-s + (−1.08 − 1.64i)6-s + (−2.26 − 1.98i)7-s + (−2.57 − 0.967i)8-s + (−1.07 + 0.402i)9-s + (3.04 − 2.66i)10-s + (4.37 + 1.20i)11-s + (−0.0737 + 0.111i)12-s + (−5.09 + 1.40i)13-s + (−1.34 + 4.15i)14-s + (1.80 + 3.34i)15-s + (0.187 + 4.18i)16-s + (1.64 + 1.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.402 − 0.941i)2-s + (0.773 − 0.140i)3-s + (−0.0339 + 0.0355i)4-s + (0.385 + 1.18i)5-s + (−0.443 − 0.671i)6-s + (−0.857 − 0.749i)7-s + (−0.911 − 0.342i)8-s + (−0.357 + 0.134i)9-s + (0.963 − 0.841i)10-s + (1.32 + 0.364i)11-s + (−0.0212 + 0.0322i)12-s + (−1.41 + 0.390i)13-s + (−0.360 + 1.10i)14-s + (0.465 + 0.864i)15-s + (0.0469 + 1.04i)16-s + (0.398 + 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799129 - 0.487398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799129 - 0.487398i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (-4.00 + 7.41i)T \) |
| good | 2 | \( 1 + (0.569 + 1.33i)T + (-1.38 + 1.44i)T^{2} \) |
| 3 | \( 1 + (-1.33 + 0.243i)T + (2.80 - 1.05i)T^{2} \) |
| 5 | \( 1 + (-0.862 - 2.65i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (2.26 + 1.98i)T + (0.939 + 6.93i)T^{2} \) |
| 11 | \( 1 + (-4.37 - 1.20i)T + (9.44 + 5.64i)T^{2} \) |
| 13 | \( 1 + (5.09 - 1.40i)T + (11.1 - 6.66i)T^{2} \) |
| 17 | \( 1 + (-1.64 - 1.19i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 2.93i)T + (-10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (-1.08 - 4.75i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.05 + 7.77i)T + (-27.9 - 7.71i)T^{2} \) |
| 31 | \( 1 + (-0.0798 + 1.77i)T + (-30.8 - 2.77i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 5.43i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (2.63 - 3.29i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.75 + 0.608i)T + (42.3 - 7.67i)T^{2} \) |
| 47 | \( 1 + (9.05 + 1.64i)T + (44.0 + 16.5i)T^{2} \) |
| 53 | \( 1 + (-0.00506 - 0.00530i)T + (-2.37 + 52.9i)T^{2} \) |
| 59 | \( 1 + (2.15 - 3.26i)T + (-23.1 - 54.2i)T^{2} \) |
| 61 | \( 1 + (2.44 - 2.13i)T + (8.18 - 60.4i)T^{2} \) |
| 67 | \( 1 + (-3.36 + 3.51i)T + (-3.00 - 66.9i)T^{2} \) |
| 73 | \( 1 + (-0.0284 - 0.0664i)T + (-50.4 + 52.7i)T^{2} \) |
| 79 | \( 1 + (-11.0 - 4.13i)T + (59.4 + 51.9i)T^{2} \) |
| 83 | \( 1 + (-2.61 + 3.95i)T + (-32.6 - 76.3i)T^{2} \) |
| 89 | \( 1 + (-7.88 - 8.24i)T + (-3.99 + 88.9i)T^{2} \) |
| 97 | \( 1 + (11.5 + 14.5i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−14.45369861605859546323840961709, −13.51554798669699934672443122039, −12.04452763543725302237210320664, −11.05626572911629380163731132079, −9.784061806058948778373735236814, −9.440277084033136979560944885293, −7.36351295365907100075017872449, −6.37543746747243414163247255099, −3.51365658431700282687931433630, −2.35835351724345313939987369306,
3.03441496665753651778186429294, 5.35820750049783639090855612973, 6.59211853227661713877202243241, 8.168971592977191115754921022755, 9.059202347972993609175397705272, 9.546970434790528762955383283705, 12.02561285940472504316875405819, 12.57034528892460498743221746256, 14.23982362396703446988332909061, 14.87290205226969924916627786209
