| L(s) = 1 | + (0.931 − 0.257i)2-s + (1.04 − 2.45i)3-s + (−0.915 + 0.546i)4-s + (−1.09 + 3.35i)5-s + (0.346 − 2.55i)6-s + (−0.0991 − 2.20i)7-s + (−2.04 + 2.14i)8-s + (−2.85 − 2.98i)9-s + (−0.153 + 3.40i)10-s + (3.81 + 0.692i)11-s + (0.382 + 2.82i)12-s + (−3.34 + 0.606i)13-s + (−0.659 − 2.03i)14-s + (7.10 + 6.20i)15-s + (−0.346 + 0.643i)16-s + (1.77 − 1.28i)17-s + ⋯ |
| L(s) = 1 | + (0.658 − 0.181i)2-s + (0.605 − 1.41i)3-s + (−0.457 + 0.273i)4-s + (−0.488 + 1.50i)5-s + (0.141 − 1.04i)6-s + (−0.0374 − 0.834i)7-s + (−0.723 + 0.757i)8-s + (−0.950 − 0.993i)9-s + (−0.0484 + 1.07i)10-s + (1.15 + 0.208i)11-s + (0.110 + 0.814i)12-s + (−0.927 + 0.168i)13-s + (−0.176 − 0.542i)14-s + (1.83 + 1.60i)15-s + (−0.0865 + 0.160i)16-s + (0.430 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13237 - 0.390746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13237 - 0.390746i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (0.559 - 8.40i)T \) |
| good | 2 | \( 1 + (-0.931 + 0.257i)T + (1.71 - 1.02i)T^{2} \) |
| 3 | \( 1 + (-1.04 + 2.45i)T + (-2.07 - 2.16i)T^{2} \) |
| 5 | \( 1 + (1.09 - 3.35i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (0.0991 + 2.20i)T + (-6.97 + 0.627i)T^{2} \) |
| 11 | \( 1 + (-3.81 - 0.692i)T + (10.2 + 3.86i)T^{2} \) |
| 13 | \( 1 + (3.34 - 0.606i)T + (12.1 - 4.56i)T^{2} \) |
| 17 | \( 1 + (-1.77 + 1.28i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.31 - 1.15i)T + (2.55 - 18.8i)T^{2} \) |
| 23 | \( 1 + (4.87 + 6.11i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-3.21 - 0.289i)T + (28.5 + 5.17i)T^{2} \) |
| 31 | \( 1 + (-1.95 - 3.62i)T + (-17.0 + 25.8i)T^{2} \) |
| 37 | \( 1 + (4.09 - 5.13i)T + (-8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + (-6.24 - 3.00i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (6.12 + 9.28i)T + (-16.9 + 39.5i)T^{2} \) |
| 47 | \( 1 + (0.119 + 0.280i)T + (-32.4 + 33.9i)T^{2} \) |
| 53 | \( 1 + (-5.79 - 3.46i)T + (25.1 + 46.6i)T^{2} \) |
| 59 | \( 1 + (-0.957 - 7.06i)T + (-56.8 + 15.6i)T^{2} \) |
| 61 | \( 1 + (0.0768 - 1.71i)T + (-60.7 - 5.46i)T^{2} \) |
| 67 | \( 1 + (-6.12 + 3.65i)T + (31.7 - 58.9i)T^{2} \) |
| 73 | \( 1 + (5.71 - 1.57i)T + (62.6 - 37.4i)T^{2} \) |
| 79 | \( 1 + (-2.62 + 2.74i)T + (-3.54 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-1.26 - 9.31i)T + (-80.0 + 22.0i)T^{2} \) |
| 89 | \( 1 + (12.1 + 7.25i)T + (42.1 + 78.3i)T^{2} \) |
| 97 | \( 1 + (10.9 - 5.25i)T + (60.4 - 75.8i)T^{2} \) |
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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
−14.20271805480320099191030599499, −13.80947355320116503270671359555, −12.35821375367478228955731762573, −11.83485664859860681969273516121, −10.20764792117703503016066365169, −8.427375342617625989217004250037, −7.28749252517965835189669377803, −6.57363899981423543762181027696, −4.02922787240330799576684516824, −2.69663430190045279102089062033,
3.76526042190777589546207758867, 4.66192514749211852168455043925, 5.66228355697958240988851661677, 8.353484732252221102991812262167, 9.220982177497088144971092508519, 9.785771507065940480016786087772, 11.80923027358380581055716585040, 12.64989258330662784092656594272, 14.00214985419884962756388948570, 14.90738501131937176947904193061
