L(s) = 1 | + (2.09 + 0.370i)2-s + (1.43 − 1.70i)3-s + (2.39 + 0.871i)4-s + (3.64 − 3.05i)6-s + (1.28 − 0.742i)7-s + (1.01 + 0.583i)8-s + (−0.342 − 1.94i)9-s + (−2.34 + 4.05i)11-s + (4.91 − 2.84i)12-s + (0.232 + 0.276i)13-s + (2.97 − 1.08i)14-s + (−1.99 − 1.67i)16-s + (−5.39 − 0.951i)17-s − 4.21i·18-s + (1.68 + 4.01i)19-s + ⋯ |
L(s) = 1 | + (1.48 + 0.261i)2-s + (0.827 − 0.986i)3-s + (1.19 + 0.435i)4-s + (1.48 − 1.24i)6-s + (0.486 − 0.280i)7-s + (0.357 + 0.206i)8-s + (−0.114 − 0.648i)9-s + (−0.705 + 1.22i)11-s + (1.42 − 0.819i)12-s + (0.0643 + 0.0767i)13-s + (0.795 − 0.289i)14-s + (−0.499 − 0.418i)16-s + (−1.30 − 0.230i)17-s − 0.992i·18-s + (0.386 + 0.922i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.65354 - 0.800742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.65354 - 0.800742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-1.68 - 4.01i)T \) |
good | 2 | \( 1 + (-2.09 - 0.370i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.43 + 1.70i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 0.742i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.34 - 4.05i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.232 - 0.276i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.39 + 0.951i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.10 + 5.79i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.155 - 0.882i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.40 + 4.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 + (4.01 + 3.36i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.46 - 6.78i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-10.7 + 1.88i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 6.12i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.70 + 9.65i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.20 + 0.803i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (8.71 - 1.53i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (6.02 - 2.19i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.83 + 2.19i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.58 + 1.32i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (5.33 - 3.08i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.54 + 2.13i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.64 - 0.819i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.32772938359826946288478959668, −10.17063216376304815905080336927, −8.884717103669695229145029870707, −7.85483526484099295629067074722, −7.14607899205650509973119825341, −6.38376753425725065272358611340, −5.01460663347781071480885088671, −4.30677364175799079632261049119, −2.87975938184845569220030307829, −1.95687923481329987959609869755,
2.43770685442395939237668758451, 3.29709555839379287180654758313, 4.17853791548410267620319650173, 5.12261436088410499622085928889, 5.88411171383339618864430657684, 7.31226787821157421002778335324, 8.798556665417347860447193440098, 8.977166388245733523112110636372, 10.58072344932625601146823818601, 11.08031097172671553496277633274