L(s) = 1 | + 0.601i·2-s + (0.926 − 2.85i)3-s + 3.63·4-s + 2.23i·5-s + (1.71 + 0.557i)6-s + 2.64·7-s + 4.59i·8-s + (−7.28 − 5.28i)9-s − 1.34·10-s − 13.9i·11-s + (3.37 − 10.3i)12-s + 14.4·13-s + 1.59i·14-s + (6.38 + 2.07i)15-s + 11.7·16-s + 26.3i·17-s + ⋯ |
L(s) = 1 | + 0.300i·2-s + (0.308 − 0.951i)3-s + 0.909·4-s + 0.447i·5-s + (0.286 + 0.0929i)6-s + 0.377·7-s + 0.574i·8-s + (−0.809 − 0.587i)9-s − 0.134·10-s − 1.26i·11-s + (0.280 − 0.864i)12-s + 1.10·13-s + 0.113i·14-s + (0.425 + 0.138i)15-s + 0.736·16-s + 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74926 - 0.276872i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.74926 - 0.276872i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.926 + 2.85i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
good | 2 | \( 1 - 0.601iT - 4T^{2} \) |
| 11 | \( 1 + 13.9iT - 121T^{2} \) |
| 13 | \( 1 - 14.4T + 169T^{2} \) |
| 17 | \( 1 - 26.3iT - 289T^{2} \) |
| 19 | \( 1 + 32.8T + 361T^{2} \) |
| 23 | \( 1 + 6.45iT - 529T^{2} \) |
| 29 | \( 1 + 2.47iT - 841T^{2} \) |
| 31 | \( 1 + 15.3T + 961T^{2} \) |
| 37 | \( 1 + 6.17T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 66.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 45.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.37iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 55.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 83.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 108.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 77.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 17.6T + 9.40e3T^{2} \) |
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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
−13.50830202906586403099416102409, −12.51441783850085889720731647535, −11.20727682861484214118898703850, −10.76928541956292278848106081596, −8.521496792550653465295608376992, −8.043670325309958994129492715495, −6.46259284591415595366551840599, −6.08758113065455140509813043278, −3.39671233495609847048447076266, −1.79364788644696297321758836854,
2.16165337837384773882818846755, 3.88476704097802341068979590876, 5.21079187235602741729178554638, 6.85586526562034026306483136334, 8.258679684707942871150579322774, 9.419254043078478706453081848979, 10.47625971924955615980622925741, 11.31679455051549946699341008402, 12.33545581245339276863109148789, 13.59608975553620305115556490548