L(s) = 1 | + (0.587 + 0.809i)2-s + (−3.21 − 1.04i)3-s + (−0.309 + 0.951i)4-s + (−2.16 − 0.563i)5-s + (−1.04 − 3.21i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (6.82 + 4.95i)9-s + (−0.815 − 2.08i)10-s + (1.59 − 1.16i)11-s + (1.98 − 2.73i)12-s + (2.90 − 3.99i)13-s + (−0.809 + 0.587i)14-s + (6.36 + 4.07i)15-s + (−0.809 − 0.587i)16-s + (3.27 − 1.06i)17-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (−1.85 − 0.603i)3-s + (−0.154 + 0.475i)4-s + (−0.967 − 0.252i)5-s + (−0.426 − 1.31i)6-s + 0.377i·7-s + (−0.336 + 0.109i)8-s + (2.27 + 1.65i)9-s + (−0.257 − 0.658i)10-s + (0.482 − 0.350i)11-s + (0.573 − 0.789i)12-s + (0.805 − 1.10i)13-s + (−0.216 + 0.157i)14-s + (1.64 + 1.05i)15-s + (−0.202 − 0.146i)16-s + (0.795 − 0.258i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.782826 + 0.0960128i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782826 + 0.0960128i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (2.16 + 0.563i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (3.21 + 1.04i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-1.59 + 1.16i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.90 + 3.99i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.27 + 1.06i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.05 - 6.33i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.525 + 0.723i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.02 + 6.23i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.540 - 1.66i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.226 - 0.311i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.25 - 5.27i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.26iT - 43T^{2} \) |
| 47 | \( 1 + (2.04 + 0.663i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.534 - 0.173i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (2.06 + 1.50i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.61 + 4.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.49 + 2.43i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-4.50 + 13.8i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.37 - 10.1i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.28 + 7.03i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.05 + 0.666i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.76 - 1.28i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (12.3 + 4.02i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.70239116084318614388408592115, −11.03275688522854181600692839901, −9.920537780429180164705571890454, −8.131718363924138959578922838438, −7.67665077229850597847989383673, −6.39395743308690791614628311860, −5.79510377048907009871359641071, −4.90986428475512388995224397062, −3.67530811164610863967879526882, −0.920584841762871446543103938644,
0.975489635796535892898117695172, 3.70116498404846069232623535031, 4.35843519424285769594219228732, 5.31408282738326909387330099687, 6.54361715182702946779054637943, 7.15847461975083973136085626882, 9.049260659658308399522933353856, 10.02371163150315731508029397033, 11.01498963609932163698988349379, 11.29256715997861551764378729358