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
−14.23188049588928558836265398109, −13.96414534705247236635470164107, −12.61684445677489373655771270629, −11.68752285616506553515980091427, −10.07854445038091162914705913289, −8.834010957936348036397938616574, −7.984369866332995480963524018726, −5.96478057340770448926495622838, −4.42164686277683334935520049222, −3.14854463579830299302408403715,
2.92038159389523541261406146752, 4.69386516995818912719307573036, 5.98398775107233111670625167529, 7.77299592373852683001503721650, 8.887448972542115033966799918589, 10.01979622268866123420458994964, 11.60642564835770306684080777374, 12.81737579776408677722619696536, 13.70881025301569554316496911936, 14.59551604445695820491073165321