| L(s) = 1 | + (−0.724 − 2.23i)2-s + (0.809 − 0.587i)3-s + (−2.83 + 2.05i)4-s + (0.884 + 2.05i)5-s + (−1.89 − 1.37i)6-s − 7-s + (2.85 + 2.07i)8-s + (0.309 − 0.951i)9-s + (3.94 − 3.46i)10-s + (−0.733 − 2.25i)11-s + (−1.08 + 3.33i)12-s + (2.06 − 6.34i)13-s + (0.724 + 2.23i)14-s + (1.92 + 1.14i)15-s + (0.389 − 1.19i)16-s + (−5.57 − 4.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.512 − 1.57i)2-s + (0.467 − 0.339i)3-s + (−1.41 + 1.02i)4-s + (0.395 + 0.918i)5-s + (−0.774 − 0.562i)6-s − 0.377·7-s + (1.00 + 0.732i)8-s + (0.103 − 0.317i)9-s + (1.24 − 1.09i)10-s + (−0.221 − 0.680i)11-s + (−0.312 + 0.961i)12-s + (0.571 − 1.75i)13-s + (0.193 + 0.596i)14-s + (0.496 + 0.294i)15-s + (0.0973 − 0.299i)16-s + (−1.35 − 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0332215 + 0.996511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0332215 + 0.996511i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.884 - 2.05i)T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + (0.724 + 2.23i)T + (-1.61 + 1.17i)T^{2} \) |
| 11 | \( 1 + (0.733 + 2.25i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.06 + 6.34i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (5.57 + 4.04i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.145 - 0.105i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.19 + 6.76i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.78 + 2.02i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.34 - 1.70i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.00316 + 0.00973i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.49 + 7.68i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 + (1.46 - 1.06i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.764 + 0.555i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.67 - 11.3i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.945 - 2.90i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-9.67 - 7.02i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (3.55 - 2.58i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.51 - 7.72i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.52 - 1.10i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.4 - 7.61i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.58 + 11.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.48 - 6.16i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.61476201321178468643865874129, −9.774216256327565622681303153611, −8.823627877215005660937993924717, −8.121037521123889110100946646866, −6.86030390922233275319429897533, −5.80825686475454980138770038428, −4.03263432963180291016533158758, −2.84097503192873371642082209737, −2.54551241029968520270470164879, −0.65469372304941525353056843319,
1.86698595454416543282763261902, 4.11778910393737390725341452396, 4.84705106751706891456714327487, 6.07903342400277803355252357674, 6.71189830964281197342920119429, 7.85229214389229484372819076648, 8.620712818404164777222585451593, 9.364751606600046956915396290759, 9.690015271014731358113453748808, 11.10918095850578167683110810189
