| L(s) = 1 | + (0.357 − 2.48i)2-s + (−0.415 − 0.909i)3-s + (−4.14 − 1.21i)4-s + (2.65 + 3.06i)5-s + (−2.41 + 0.708i)6-s + (−1.15 − 0.744i)7-s + (−2.41 + 5.29i)8-s + (−0.654 + 0.755i)9-s + (8.57 − 5.51i)10-s + (−0.00388 − 0.0270i)11-s + (0.614 + 4.27i)12-s + (−0.527 + 0.339i)13-s + (−2.26 + 2.61i)14-s + (1.68 − 3.68i)15-s + (5.04 + 3.24i)16-s + (4.13 − 1.21i)17-s + ⋯ |
| L(s) = 1 | + (0.252 − 1.75i)2-s + (−0.239 − 0.525i)3-s + (−2.07 − 0.608i)4-s + (1.18 + 1.37i)5-s + (−0.984 + 0.289i)6-s + (−0.437 − 0.281i)7-s + (−0.855 + 1.87i)8-s + (−0.218 + 0.251i)9-s + (2.71 − 1.74i)10-s + (−0.00117 − 0.00814i)11-s + (0.177 + 1.23i)12-s + (−0.146 + 0.0940i)13-s + (−0.605 + 0.699i)14-s + (0.434 − 0.952i)15-s + (1.26 + 0.811i)16-s + (1.00 − 0.294i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.455386 - 0.847738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455386 - 0.847738i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-1.11 - 4.66i)T \) |
| good | 2 | \( 1 + (-0.357 + 2.48i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-2.65 - 3.06i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.15 + 0.744i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.00388 + 0.0270i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.527 - 0.339i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 1.21i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (4.79 + 1.40i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (3.54 - 1.04i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.200 - 0.438i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.44 + 1.66i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (6.56 + 7.57i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.36 + 2.98i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 8.98T + 47T^{2} \) |
| 53 | \( 1 + (-1.54 - 0.993i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (3.51 - 2.25i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.92 + 4.20i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.630 - 4.38i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.02 + 7.15i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (5.72 + 1.68i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.26 + 0.815i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.61 + 6.47i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.53 - 12.1i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (1.41 + 1.63i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−13.81983037085666655730390253293, −13.26286582497314835339197542608, −12.09648412693333791534261937638, −10.92621106147402566503850479687, −10.27642311071601384791173047257, −9.309431876606769252560371652490, −7.03133056147888292007657999388, −5.58770271701022535878679679935, −3.37282705052312990836249607613, −2.06804246159223874210557656941,
4.50148315843682922811825352415, 5.55893831959125006122150709896, 6.32413086973817476837253273370, 8.219636143715080311565094971812, 9.110411132293069162227463569972, 10.05004124890542602920608607187, 12.48330338645854550340218988704, 13.16425456362938904631195335022, 14.30637438326722459774532966889, 15.23415870776629258043628980528
