| L(s) = 1 | + (1.70 + 1.89i)2-s + (−0.375 − 0.167i)3-s + (−0.471 + 4.48i)4-s + (−3.35 − 0.712i)5-s + (−0.323 − 0.996i)6-s + (−5.18 + 3.76i)8-s + (−1.89 − 2.10i)9-s + (−4.37 − 7.57i)10-s + (−2.85 + 1.68i)11-s + (0.926 − 1.60i)12-s + (−0.672 + 2.06i)13-s + (1.13 + 0.827i)15-s + (−7.17 − 1.52i)16-s + (−3.02 + 3.35i)17-s + (0.755 − 7.18i)18-s + (0.253 + 2.41i)19-s + ⋯ |
| L(s) = 1 | + (1.20 + 1.34i)2-s + (−0.216 − 0.0964i)3-s + (−0.235 + 2.24i)4-s + (−1.49 − 0.318i)5-s + (−0.132 − 0.406i)6-s + (−1.83 + 1.33i)8-s + (−0.631 − 0.701i)9-s + (−1.38 − 2.39i)10-s + (−0.861 + 0.507i)11-s + (0.267 − 0.463i)12-s + (−0.186 + 0.573i)13-s + (0.294 + 0.213i)15-s + (−1.79 − 0.381i)16-s + (−0.733 + 0.814i)17-s + (0.178 − 1.69i)18-s + (0.0581 + 0.553i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.336300 - 0.835017i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.336300 - 0.835017i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (2.85 - 1.68i)T \) |
| good | 2 | \( 1 + (-1.70 - 1.89i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (0.375 + 0.167i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (3.35 + 0.712i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (0.672 - 2.06i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.02 - 3.35i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.253 - 2.41i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.324 - 0.561i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.01 - 0.736i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.86 + 1.67i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-4.57 + 2.03i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 1.54i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + (-0.526 - 5.01i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (13.1 - 2.78i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.801 - 7.62i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (14.0 + 2.98i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (3.11 + 5.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.30 - 4.02i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.619 + 5.89i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-6.53 - 7.25i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (2.58 + 7.96i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.38 - 4.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.69 - 8.27i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.85966371327760044010953429280, −10.84575212860898449556928841937, −9.174232517335082107408432376217, −8.116588058396734849200037855670, −7.74838773371647703836856910597, −6.68872054976625357885987887721, −5.93182729451314082109280606939, −4.70055460921113812929618343724, −4.17070894210957072320009959871, −3.06770776177842458401754346667,
0.34757693611106530121372578341, 2.64654342089174367538011951872, 3.18355818523895266659041318530, 4.52676387973645801991421793733, 5.00742466120071647052574346368, 6.23193730032051155412942201381, 7.62181405778240404231097175978, 8.469865929416593440102866247825, 9.927394501861952614373356151207, 10.85903016166162539368445287810
