| L(s) = 1 | + (0.365 + 0.328i)2-s + (−0.994 + 0.104i)3-s + (−0.183 − 1.74i)4-s + (2.22 − 0.252i)5-s + (−0.397 − 0.288i)6-s + (2.15 − 1.53i)7-s + (1.08 − 1.49i)8-s + (0.978 − 0.207i)9-s + (0.894 + 0.638i)10-s + (−1.35 − 0.288i)11-s + (0.365 + 1.72i)12-s + (−5.24 − 1.70i)13-s + (1.29 + 0.145i)14-s + (−2.18 + 0.483i)15-s + (−2.55 + 0.542i)16-s + (−0.131 − 0.295i)17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.232i)2-s + (−0.574 + 0.0603i)3-s + (−0.0919 − 0.874i)4-s + (0.993 − 0.113i)5-s + (−0.162 − 0.117i)6-s + (0.813 − 0.581i)7-s + (0.383 − 0.528i)8-s + (0.326 − 0.0693i)9-s + (0.282 + 0.201i)10-s + (−0.409 − 0.0869i)11-s + (0.105 + 0.496i)12-s + (−1.45 − 0.472i)13-s + (0.345 + 0.0388i)14-s + (−0.563 + 0.124i)15-s + (−0.638 + 0.135i)16-s + (−0.0318 − 0.0715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31674 - 0.846454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31674 - 0.846454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (-2.22 + 0.252i)T \) |
| 7 | \( 1 + (-2.15 + 1.53i)T \) |
| good | 2 | \( 1 + (-0.365 - 0.328i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (1.35 + 0.288i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (5.24 + 1.70i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.131 + 0.295i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.000216 + 0.00206i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-2.00 - 1.80i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-7.74 + 5.62i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.46 + 0.653i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.215 + 1.01i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (-1.62 + 5.00i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 2.70iT - 43T^{2} \) |
| 47 | \( 1 + (1.90 - 4.28i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (1.09 - 0.115i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-4.86 - 5.40i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (8.26 - 9.18i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.12 + 4.77i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-3.29 + 2.39i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.75 - 12.9i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-10.5 - 4.67i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.98 - 4.10i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-8.58 + 9.53i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-8.85 - 12.1i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.39821632310292581505590318011, −10.14795184458341781316561198826, −9.187260725950848455615160461400, −7.79819780136227900414101109141, −6.86825179816928319693981006623, −5.87903571593491648966800659637, −5.08772957724994181276333669117, −4.52304664862421662604918145521, −2.38389776069313093728599614917, −0.954834646332855145790429625187,
1.95664191885964544920363654592, 2.89524511843727700414135590817, 4.78387489544633822085895704168, 5.02724702053522349170631273138, 6.43286356842542695724486843382, 7.34829055735518191553481920186, 8.364441978103206976722481613507, 9.270913418208457093066603501336, 10.28891778599583454767596895363, 11.11169934061662395661360163501
