L(s) = 1 | + (2.58 + 0.455i)2-s + (−0.597 + 0.711i)3-s + (4.58 + 1.66i)4-s + (−1.86 + 1.56i)6-s + (−3.31 + 1.91i)7-s + (6.53 + 3.77i)8-s + (0.371 + 2.10i)9-s + (1.63 − 2.82i)11-s + (−3.92 + 2.26i)12-s + (0.234 + 0.278i)13-s + (−9.44 + 3.43i)14-s + (7.67 + 6.44i)16-s + (−0.673 − 0.118i)17-s + 5.60i·18-s + (3.86 − 2.01i)19-s + ⋯ |
L(s) = 1 | + (1.82 + 0.321i)2-s + (−0.344 + 0.410i)3-s + (2.29 + 0.833i)4-s + (−0.761 + 0.639i)6-s + (−1.25 + 0.724i)7-s + (2.30 + 1.33i)8-s + (0.123 + 0.701i)9-s + (0.492 − 0.853i)11-s + (−1.13 + 0.653i)12-s + (0.0649 + 0.0773i)13-s + (−2.52 + 0.918i)14-s + (1.91 + 1.61i)16-s + (−0.163 − 0.0288i)17-s + 1.32i·18-s + (0.886 − 0.461i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.77216 + 1.94241i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.77216 + 1.94241i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.86 + 2.01i)T \) |
good | 2 | \( 1 + (-2.58 - 0.455i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (0.597 - 0.711i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (3.31 - 1.91i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.234 - 0.278i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.673 + 0.118i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-3.20 + 8.80i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.267 + 1.51i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 2.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.163iT - 37T^{2} \) |
| 41 | \( 1 + (5.64 + 4.73i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.885 - 2.43i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (8.66 - 1.52i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.80 - 10.4i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.35 + 7.67i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.27 + 0.827i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-9.41 + 1.65i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.81 + 1.75i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (5.06 - 6.03i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.14 - 5.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.631 - 0.364i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.68 - 8.12i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (7.43 + 1.31i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.40277066692953322954693939621, −10.72634493540076102549548090825, −9.516988340588945016766392316225, −8.298121539032955488688063383774, −6.89454700408789703877304055094, −6.29316842968618124972426267256, −5.42544017742044913922165185783, −4.59670051193064701828409452563, −3.42211990345161623604490137846, −2.58368945773848486611364172962,
1.48465478429656391854419743031, 3.27933623263074713969021179387, 3.77963258781365940106964360764, 5.06228292680674714044247082061, 6.09430953116259447690943857675, 6.81767768020881153729527246069, 7.37114146613426814029338565309, 9.521002349709157074810424559650, 10.09133213138965391329790755776, 11.39759632397158543728465977905