| 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
−11.29256715997861551764378729358, −11.01498963609932163698988349379, −10.02371163150315731508029397033, −9.049260659658308399522933353856, −7.15847461975083973136085626882, −6.54361715182702946779054637943, −5.31408282738326909387330099687, −4.35843519424285769594219228732, −3.70116498404846069232623535031, −0.975489635796535892898117695172,
0.920584841762871446543103938644, 3.67530811164610863967879526882, 4.90986428475512388995224397062, 5.79510377048907009871359641071, 6.39395743308690791614628311860, 7.67665077229850597847989383673, 8.131718363924138959578922838438, 9.920537780429180164705571890454, 11.03275688522854181600692839901, 11.70239116084318614388408592115
