| L(s) = 1 | + (0.0758 − 0.177i)2-s + (−0.355 − 0.0645i)3-s + (1.35 + 1.41i)4-s + (−0.0139 + 0.0430i)5-s + (−0.0384 + 0.0581i)6-s + (1.74 − 1.52i)7-s + (0.715 − 0.268i)8-s + (−2.68 − 1.00i)9-s + (0.00658 + 0.00574i)10-s + (−3.34 + 0.923i)11-s + (−0.390 − 0.592i)12-s + (0.0252 + 0.00696i)13-s + (−0.137 − 0.424i)14-s + (0.00775 − 0.0144i)15-s + (−0.169 + 3.77i)16-s + (−2.92 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.0536 − 0.125i)2-s + (−0.205 − 0.0372i)3-s + (0.678 + 0.709i)4-s + (−0.00625 + 0.0192i)5-s + (−0.0156 + 0.0237i)6-s + (0.658 − 0.575i)7-s + (0.253 − 0.0949i)8-s + (−0.895 − 0.336i)9-s + (0.00208 + 0.00181i)10-s + (−1.00 + 0.278i)11-s + (−0.112 − 0.170i)12-s + (0.00699 + 0.00193i)13-s + (−0.0368 − 0.113i)14-s + (0.00200 − 0.00372i)15-s + (−0.0423 + 0.943i)16-s + (−0.710 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975571 + 0.0310057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975571 + 0.0310057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (4.30 - 7.24i)T \) |
| good | 2 | \( 1 + (-0.0758 + 0.177i)T + (-1.38 - 1.44i)T^{2} \) |
| 3 | \( 1 + (0.355 + 0.0645i)T + (2.80 + 1.05i)T^{2} \) |
| 5 | \( 1 + (0.0139 - 0.0430i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.74 + 1.52i)T + (0.939 - 6.93i)T^{2} \) |
| 11 | \( 1 + (3.34 - 0.923i)T + (9.44 - 5.64i)T^{2} \) |
| 13 | \( 1 + (-0.0252 - 0.00696i)T + (11.1 + 6.66i)T^{2} \) |
| 17 | \( 1 + (2.92 - 2.12i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.60 + 4.83i)T + (-10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (-0.970 + 4.24i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (0.203 + 1.50i)T + (-27.9 + 7.71i)T^{2} \) |
| 31 | \( 1 + (0.112 + 2.51i)T + (-30.8 + 2.77i)T^{2} \) |
| 37 | \( 1 + (-1.86 - 8.18i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-2.11 - 2.65i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-2.59 - 0.233i)T + (42.3 + 7.67i)T^{2} \) |
| 47 | \( 1 + (-11.6 + 2.12i)T + (44.0 - 16.5i)T^{2} \) |
| 53 | \( 1 + (5.80 - 6.07i)T + (-2.37 - 52.9i)T^{2} \) |
| 59 | \( 1 + (4.95 + 7.51i)T + (-23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (5.89 + 5.15i)T + (8.18 + 60.4i)T^{2} \) |
| 67 | \( 1 + (-3.75 - 3.92i)T + (-3.00 + 66.9i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 6.00i)T + (-50.4 - 52.7i)T^{2} \) |
| 79 | \( 1 + (1.04 - 0.390i)T + (59.4 - 51.9i)T^{2} \) |
| 83 | \( 1 + (1.50 + 2.27i)T + (-32.6 + 76.3i)T^{2} \) |
| 89 | \( 1 + (-7.84 + 8.20i)T + (-3.99 - 88.9i)T^{2} \) |
| 97 | \( 1 + (-5.45 + 6.84i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−14.82280502334286894572061031210, −13.41293839572185818681466719903, −12.49108107405981275929293729741, −11.20171980207340713497263272344, −10.72727391063428073099621477368, −8.741459678387590722716256453624, −7.66728628307434185213961861738, −6.42378020109419530649450948408, −4.59500504455092300549716497451, −2.69010302283295469022832294942,
2.41933572842501425598137893375, 5.11550844359055600561844694412, 5.98169197811762915200857182034, 7.61657387890337517046742669258, 8.869704292559714050352072339069, 10.55297538240439066332908571749, 11.16289466511086383282211900858, 12.30664337645824629146813602140, 13.86581378151500875401382304939, 14.70768962440082112598773268701
