| L(s) = 1 | + (0.871 − 2.03i)2-s + (0.685 + 0.124i)3-s + (−2.01 − 2.10i)4-s + (−0.768 + 2.36i)5-s + (0.850 − 1.28i)6-s + (−1.65 + 1.45i)7-s + (−1.89 + 0.711i)8-s + (−2.35 − 0.883i)9-s + (4.14 + 3.62i)10-s + (0.236 − 0.0652i)11-s + (−1.11 − 1.69i)12-s + (−0.742 − 0.204i)13-s + (1.50 + 4.64i)14-s + (−0.820 + 1.52i)15-s + (0.0600 − 1.33i)16-s + (6.29 − 4.57i)17-s + ⋯ |
| L(s) = 1 | + (0.616 − 1.44i)2-s + (0.395 + 0.0718i)3-s + (−1.00 − 1.05i)4-s + (−0.343 + 1.05i)5-s + (0.347 − 0.526i)6-s + (−0.627 + 0.548i)7-s + (−0.670 + 0.251i)8-s + (−0.784 − 0.294i)9-s + (1.31 + 1.14i)10-s + (0.0713 − 0.0196i)11-s + (−0.322 − 0.489i)12-s + (−0.206 − 0.0568i)13-s + (0.403 + 1.24i)14-s + (−0.211 + 0.393i)15-s + (0.0150 − 0.334i)16-s + (1.52 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.922711 - 0.744990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.922711 - 0.744990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (7.80 + 3.18i)T \) |
| good | 2 | \( 1 + (-0.871 + 2.03i)T + (-1.38 - 1.44i)T^{2} \) |
| 3 | \( 1 + (-0.685 - 0.124i)T + (2.80 + 1.05i)T^{2} \) |
| 5 | \( 1 + (0.768 - 2.36i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (1.65 - 1.45i)T + (0.939 - 6.93i)T^{2} \) |
| 11 | \( 1 + (-0.236 + 0.0652i)T + (9.44 - 5.64i)T^{2} \) |
| 13 | \( 1 + (0.742 + 0.204i)T + (11.1 + 6.66i)T^{2} \) |
| 17 | \( 1 + (-6.29 + 4.57i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.932 + 1.73i)T + (-10.4 + 15.8i)T^{2} \) |
| 23 | \( 1 + (1.49 - 6.55i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 8.21i)T + (-27.9 + 7.71i)T^{2} \) |
| 31 | \( 1 + (0.336 + 7.49i)T + (-30.8 + 2.77i)T^{2} \) |
| 37 | \( 1 + (0.840 + 3.68i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-2.72 - 3.41i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (10.1 + 0.915i)T + (42.3 + 7.67i)T^{2} \) |
| 47 | \( 1 + (-2.78 + 0.504i)T + (44.0 - 16.5i)T^{2} \) |
| 53 | \( 1 + (-2.93 + 3.06i)T + (-2.37 - 52.9i)T^{2} \) |
| 59 | \( 1 + (-6.29 - 9.53i)T + (-23.1 + 54.2i)T^{2} \) |
| 61 | \( 1 + (-8.64 - 7.55i)T + (8.18 + 60.4i)T^{2} \) |
| 67 | \( 1 + (1.97 + 2.06i)T + (-3.00 + 66.9i)T^{2} \) |
| 73 | \( 1 + (-0.820 + 1.91i)T + (-50.4 - 52.7i)T^{2} \) |
| 79 | \( 1 + (-2.55 + 0.958i)T + (59.4 - 51.9i)T^{2} \) |
| 83 | \( 1 + (4.34 + 6.57i)T + (-32.6 + 76.3i)T^{2} \) |
| 89 | \( 1 + (-8.90 + 9.31i)T + (-3.99 - 88.9i)T^{2} \) |
| 97 | \( 1 + (-0.923 + 1.15i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−14.30393659473776602511209393184, −13.26183914816578088339443215816, −11.97104081183823014484550643151, −11.43724878385387205036144555421, −10.16216007666893055689758917908, −9.225786061804918719791247222986, −7.35886509016054763442990234116, −5.54986781268706647676850003805, −3.47666943163393249263109450514, −2.79076349461045928980934240687,
3.86102612033339527346155613269, 5.24074594935129708959396267437, 6.48359615763420642807664244250, 7.995868599104524621379453975831, 8.510121432005121289545692391424, 10.21419550761606191257084020966, 12.19962248655349987212520119070, 13.05288326458809980574829185494, 14.09700075308011505440958098347, 14.80846814558602979915525273649
