| L(s) = 1 | + (1.52 − 0.137i)2-s + (0.0564 − 0.416i)3-s + (0.340 − 0.0617i)4-s + (−0.536 − 0.389i)5-s + (0.0288 − 0.643i)6-s + (−2.10 + 1.25i)7-s + (−2.44 + 0.674i)8-s + (2.72 + 0.751i)9-s + (−0.872 − 0.521i)10-s + (0.0584 − 0.0885i)11-s + (−0.00652 − 0.145i)12-s + (−1.01 − 1.54i)13-s + (−3.03 + 2.20i)14-s + (−0.192 + 0.201i)15-s + (−4.28 + 1.60i)16-s + (1.30 − 4.00i)17-s + ⋯ |
| L(s) = 1 | + (1.07 − 0.0970i)2-s + (0.0325 − 0.240i)3-s + (0.170 − 0.0308i)4-s + (−0.240 − 0.174i)5-s + (0.0117 − 0.262i)6-s + (−0.794 + 0.474i)7-s + (−0.863 + 0.238i)8-s + (0.907 + 0.250i)9-s + (−0.275 − 0.164i)10-s + (0.0176 − 0.0267i)11-s + (−0.00188 − 0.0419i)12-s + (−0.282 − 0.427i)13-s + (−0.810 + 0.589i)14-s + (−0.0497 + 0.0520i)15-s + (−1.07 + 0.401i)16-s + (0.315 − 0.972i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28805 - 0.136418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28805 - 0.136418i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (8.30 - 1.44i)T \) |
| good | 2 | \( 1 + (-1.52 + 0.137i)T + (1.96 - 0.357i)T^{2} \) |
| 3 | \( 1 + (-0.0564 + 0.416i)T + (-2.89 - 0.798i)T^{2} \) |
| 5 | \( 1 + (0.536 + 0.389i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (2.10 - 1.25i)T + (3.31 - 6.16i)T^{2} \) |
| 11 | \( 1 + (-0.0584 + 0.0885i)T + (-4.32 - 10.1i)T^{2} \) |
| 13 | \( 1 + (1.01 + 1.54i)T + (-5.10 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 4.00i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.24 - 2.34i)T + (-0.852 + 18.9i)T^{2} \) |
| 23 | \( 1 + (0.0361 + 0.158i)T + (-20.7 + 9.97i)T^{2} \) |
| 29 | \( 1 + (-3.68 - 6.85i)T + (-15.9 + 24.2i)T^{2} \) |
| 31 | \( 1 + (2.27 + 0.853i)T + (23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (0.928 - 4.06i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-6.90 + 8.65i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (2.90 - 2.54i)T + (5.77 - 42.6i)T^{2} \) |
| 47 | \( 1 + (-0.229 - 1.69i)T + (-45.3 + 12.5i)T^{2} \) |
| 53 | \( 1 + (-6.43 - 1.16i)T + (49.6 + 18.6i)T^{2} \) |
| 59 | \( 1 + (-0.471 - 10.5i)T + (-58.7 + 5.28i)T^{2} \) |
| 61 | \( 1 + (12.0 + 7.22i)T + (28.9 + 53.7i)T^{2} \) |
| 67 | \( 1 + (11.5 - 2.10i)T + (62.7 - 23.5i)T^{2} \) |
| 73 | \( 1 + (6.36 - 0.572i)T + (71.8 - 13.0i)T^{2} \) |
| 79 | \( 1 + (9.71 - 2.68i)T + (67.8 - 40.5i)T^{2} \) |
| 83 | \( 1 + (0.408 + 9.10i)T + (-82.6 + 7.44i)T^{2} \) |
| 89 | \( 1 + (-13.0 - 2.36i)T + (83.3 + 31.2i)T^{2} \) |
| 97 | \( 1 + (-0.0437 - 0.0549i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−14.40983997808146760041962027203, −13.45217797303613783932510809526, −12.51835235226189885799063473665, −11.98011300907014540050440621730, −10.18861920951883514933220051283, −8.969096056862790141551471761217, −7.36832659910208714616120503413, −5.90103645059883349398802694541, −4.59226649854614718456351627772, −3.04267626105778726816319366664,
3.46385581346051263769017061319, 4.49292996029221771114562354859, 6.12670695341345386590784498660, 7.30550598209429452540359215902, 9.239975254284571123352324834811, 10.19359305288312234158614486056, 11.75339473010624609058804994248, 12.83901719776506836451669680769, 13.51263576414881414904115883429, 14.71975469867405999078432157863
