| L(s) = 1 | + (1.46 + 1.53i)2-s + (−0.822 − 0.308i)3-s + (−0.112 + 2.50i)4-s + (−0.577 − 0.419i)5-s + (−0.732 − 1.71i)6-s + (−0.0309 + 0.228i)7-s + (−0.803 + 0.702i)8-s + (−1.67 − 1.46i)9-s + (−0.203 − 1.50i)10-s + (1.07 − 0.643i)11-s + (0.863 − 2.02i)12-s + (−2.30 − 1.37i)13-s + (−0.396 + 0.288i)14-s + (0.345 + 0.523i)15-s + (2.73 + 0.245i)16-s + (−0.509 + 1.56i)17-s + ⋯ |
| L(s) = 1 | + (1.03 + 1.08i)2-s + (−0.474 − 0.178i)3-s + (−0.0561 + 1.25i)4-s + (−0.258 − 0.187i)5-s + (−0.299 − 0.699i)6-s + (−0.0117 + 0.0864i)7-s + (−0.284 + 0.248i)8-s + (−0.559 − 0.488i)9-s + (−0.0643 − 0.474i)10-s + (0.324 − 0.194i)11-s + (0.249 − 0.583i)12-s + (−0.638 − 0.381i)13-s + (−0.105 + 0.0769i)14-s + (0.0892 + 0.135i)15-s + (0.682 + 0.0614i)16-s + (−0.123 + 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05584 + 0.631025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05584 + 0.631025i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (3.31 + 7.74i)T \) |
| good | 2 | \( 1 + (-1.46 - 1.53i)T + (-0.0897 + 1.99i)T^{2} \) |
| 3 | \( 1 + (0.822 + 0.308i)T + (2.25 + 1.97i)T^{2} \) |
| 5 | \( 1 + (0.577 + 0.419i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (0.0309 - 0.228i)T + (-6.74 - 1.86i)T^{2} \) |
| 11 | \( 1 + (-1.07 + 0.643i)T + (5.21 - 9.68i)T^{2} \) |
| 13 | \( 1 + (2.30 + 1.37i)T + (6.16 + 11.4i)T^{2} \) |
| 17 | \( 1 + (0.509 - 1.56i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (3.60 - 5.45i)T + (-7.46 - 17.4i)T^{2} \) |
| 23 | \( 1 + (-5.83 - 2.81i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.799 + 0.220i)T + (24.8 - 14.8i)T^{2} \) |
| 31 | \( 1 + (7.67 - 0.690i)T + (30.5 - 5.53i)T^{2} \) |
| 37 | \( 1 + (7.12 - 3.42i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-1.54 + 6.78i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-5.90 - 1.07i)T + (40.2 + 15.1i)T^{2} \) |
| 47 | \( 1 + (-8.01 + 3.00i)T + (35.3 - 30.9i)T^{2} \) |
| 53 | \( 1 + (0.0866 + 1.92i)T + (-52.7 + 4.75i)T^{2} \) |
| 59 | \( 1 + (1.25 - 2.92i)T + (-40.7 - 42.6i)T^{2} \) |
| 61 | \( 1 + (1.31 + 9.68i)T + (-58.8 + 16.2i)T^{2} \) |
| 67 | \( 1 + (-0.161 + 3.60i)T + (-66.7 - 6.00i)T^{2} \) |
| 73 | \( 1 + (-4.54 - 4.75i)T + (-3.27 + 72.9i)T^{2} \) |
| 79 | \( 1 + (4.26 - 3.72i)T + (10.6 - 78.2i)T^{2} \) |
| 83 | \( 1 + (-1.78 + 4.18i)T + (-57.3 - 59.9i)T^{2} \) |
| 89 | \( 1 + (0.505 + 11.2i)T + (-88.6 + 7.97i)T^{2} \) |
| 97 | \( 1 + (2.36 + 10.3i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−14.83084536126889242171875412731, −14.06405493521524225143003818529, −12.69462104314826432028765474785, −12.11844163165223145765059721797, −10.60403722250991531362924582015, −8.786465670189589168950691496169, −7.43537767633326882425128004375, −6.24182126920260817828300078792, −5.30744105697773308365273138540, −3.76444117069917880690421196390,
2.60298646792661262992627362504, 4.31984330494598018692129046884, 5.38892624920721952854722583206, 7.15848160724667784453059026024, 9.081418359336778296580779664964, 10.70704871037898283296851896783, 11.21540767019026630822753331046, 12.22622687469272261059048198846, 13.23113824469580173766638365747, 14.26910784527885144484638515567
