| L(s) = 1 | + (−0.358 − 0.933i)2-s + (−0.566 − 0.872i)3-s + (−0.743 + 0.669i)4-s + (−1.50 − 1.65i)5-s + (−0.611 + 0.841i)6-s + (1.01 + 2.44i)7-s + (0.891 + 0.453i)8-s + (0.779 − 1.75i)9-s + (−1.00 + 1.99i)10-s + (0.157 − 3.31i)11-s + (1.00 + 0.269i)12-s + (−4.88 − 0.773i)13-s + (1.91 − 1.82i)14-s + (−0.590 + 2.25i)15-s + (0.104 − 0.994i)16-s + (0.697 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.660i)2-s + (−0.327 − 0.503i)3-s + (−0.371 + 0.334i)4-s + (−0.673 − 0.739i)5-s + (−0.249 + 0.343i)6-s + (0.384 + 0.923i)7-s + (0.315 + 0.160i)8-s + (0.259 − 0.583i)9-s + (−0.317 + 0.631i)10-s + (0.0475 − 0.998i)11-s + (0.290 + 0.0777i)12-s + (−1.35 − 0.214i)13-s + (0.512 − 0.487i)14-s + (−0.152 + 0.581i)15-s + (0.0261 − 0.248i)16-s + (0.169 − 0.440i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134605 + 0.236848i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134605 + 0.236848i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 + 0.933i)T \) |
| 5 | \( 1 + (1.50 + 1.65i)T \) |
| 7 | \( 1 + (-1.01 - 2.44i)T \) |
| 11 | \( 1 + (-0.157 + 3.31i)T \) |
| good | 3 | \( 1 + (0.566 + 0.872i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (4.88 + 0.773i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.697 + 1.81i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (2.50 - 2.77i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.56 - 0.687i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (10.1 + 3.29i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.07 - 0.638i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-8.68 - 5.63i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-2.76 + 0.898i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.147 + 0.147i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.604 - 11.5i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (3.86 + 3.12i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-5.69 - 6.32i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (9.95 + 1.04i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.37 + 0.635i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (9.56 + 6.94i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.430 + 8.21i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-4.06 + 9.13i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.627 + 3.96i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-2.14 - 3.71i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.89 - 11.9i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−9.386873131787597465341438008753, −9.216726909361961148024150705769, −7.973966372923441381165328860216, −7.52470020820721258549702578595, −6.06106885377770888922910587996, −5.22767580125523006542116825494, −4.14528040373832771040085914152, −2.95537632095926435664100769892, −1.55704733691114272990527686080, −0.15629229493709204496452370799,
2.13315559299814417416239260556, 3.95029587586651010767499189166, 4.54721127339879491609200734013, 5.46487464323912779160951720055, 7.02757784651291176964033989654, 7.24467958899263935744699226784, 7.981943596151037998759548377486, 9.352932358269425242260585509990, 10.03139923566226140770741985988, 10.85692811177088198909395806151
