| L(s) = 1 | + (0.855 + 0.155i)2-s + (1.08 − 0.299i)3-s + (−1.16 − 0.437i)4-s + (−0.0956 + 0.294i)5-s + (0.973 − 0.0875i)6-s + (0.492 + 0.914i)7-s + (−2.42 − 1.44i)8-s + (−1.49 + 0.890i)9-s + (−0.127 + 0.237i)10-s + (−0.439 + 1.02i)11-s + (−1.39 − 0.125i)12-s + (−0.459 − 1.07i)13-s + (0.279 + 0.858i)14-s + (−0.0156 + 0.347i)15-s + (0.0292 + 0.0255i)16-s + (2.35 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (0.604 + 0.109i)2-s + (0.625 − 0.172i)3-s + (−0.582 − 0.218i)4-s + (−0.0427 + 0.131i)5-s + (0.397 − 0.0357i)6-s + (0.186 + 0.345i)7-s + (−0.855 − 0.511i)8-s + (−0.496 + 0.296i)9-s + (−0.0403 + 0.0749i)10-s + (−0.132 + 0.309i)11-s + (−0.402 − 0.0362i)12-s + (−0.127 − 0.298i)13-s + (0.0745 + 0.229i)14-s + (−0.00403 + 0.0898i)15-s + (0.00731 + 0.00639i)16-s + (0.572 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19010 - 0.0170464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19010 - 0.0170464i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + (8.33 + 1.23i)T \) |
| good | 2 | \( 1 + (-0.855 - 0.155i)T + (1.87 + 0.702i)T^{2} \) |
| 3 | \( 1 + (-1.08 + 0.299i)T + (2.57 - 1.53i)T^{2} \) |
| 5 | \( 1 + (0.0956 - 0.294i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.492 - 0.914i)T + (-3.85 + 5.84i)T^{2} \) |
| 11 | \( 1 + (0.439 - 1.02i)T + (-7.60 - 7.95i)T^{2} \) |
| 13 | \( 1 + (0.459 + 1.07i)T + (-8.98 + 9.39i)T^{2} \) |
| 17 | \( 1 + (-2.35 + 1.71i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0701 + 1.56i)T + (-18.9 + 1.70i)T^{2} \) |
| 23 | \( 1 + (4.59 + 2.21i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.540 - 0.819i)T + (-11.3 + 26.6i)T^{2} \) |
| 31 | \( 1 + (-6.20 + 5.41i)T + (4.16 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-7.16 + 3.44i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (2.17 - 9.52i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.0638 + 0.471i)T + (-41.4 + 11.4i)T^{2} \) |
| 47 | \( 1 + (2.69 + 0.744i)T + (40.3 + 24.1i)T^{2} \) |
| 53 | \( 1 + (4.16 - 1.56i)T + (39.9 - 34.8i)T^{2} \) |
| 59 | \( 1 + (6.33 + 0.569i)T + (58.0 + 10.5i)T^{2} \) |
| 61 | \( 1 + (3.68 - 6.84i)T + (-33.6 - 50.9i)T^{2} \) |
| 67 | \( 1 + (7.92 + 2.97i)T + (50.4 + 44.0i)T^{2} \) |
| 73 | \( 1 + (-5.30 - 0.963i)T + (68.3 + 25.6i)T^{2} \) |
| 79 | \( 1 + (10.9 + 6.54i)T + (37.4 + 69.5i)T^{2} \) |
| 83 | \( 1 + (4.71 + 0.424i)T + (81.6 + 14.8i)T^{2} \) |
| 89 | \( 1 + (4.50 - 1.69i)T + (67.0 - 58.5i)T^{2} \) |
| 97 | \( 1 + (-4.04 - 17.7i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−14.59551604445695820491073165321, −13.70881025301569554316496911936, −12.81737579776408677722619696536, −11.60642564835770306684080777374, −10.01979622268866123420458994964, −8.887448972542115033966799918589, −7.77299592373852683001503721650, −5.98398775107233111670625167529, −4.69386516995818912719307573036, −2.92038159389523541261406146752,
3.14854463579830299302408403715, 4.42164686277683334935520049222, 5.96478057340770448926495622838, 7.984369866332995480963524018726, 8.834010957936348036397938616574, 10.07854445038091162914705913289, 11.68752285616506553515980091427, 12.61684445677489373655771270629, 13.96414534705247236635470164107, 14.23188049588928558836265398109
