| L(s) = 1 | + (−0.0871 + 0.996i)2-s + (0.794 + 1.70i)3-s + (−0.984 − 0.173i)4-s + (−1.76 + 0.642i)6-s + (−3.76 + 1.00i)7-s + (0.258 − 0.965i)8-s + (−0.342 + 0.407i)9-s + (−1.56 − 2.71i)11-s + (−0.486 − 1.81i)12-s + (−1.46 − 0.684i)13-s + (−0.676 − 3.83i)14-s + (0.939 + 0.342i)16-s + (−6.30 − 0.551i)17-s + (−0.376 − 0.376i)18-s + (2.70 − 3.41i)19-s + ⋯ |
| L(s) = 1 | + (−0.0616 + 0.704i)2-s + (0.458 + 0.983i)3-s + (−0.492 − 0.0868i)4-s + (−0.720 + 0.262i)6-s + (−1.42 + 0.380i)7-s + (0.0915 − 0.341i)8-s + (−0.114 + 0.135i)9-s + (−0.472 − 0.817i)11-s + (−0.140 − 0.524i)12-s + (−0.407 − 0.189i)13-s + (−0.180 − 1.02i)14-s + (0.234 + 0.0855i)16-s + (−1.52 − 0.133i)17-s + (−0.0886 − 0.0886i)18-s + (0.621 − 0.783i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105915 - 0.0849779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105915 - 0.0849779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0871 - 0.996i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.70 + 3.41i)T \) |
| good | 3 | \( 1 + (-0.794 - 1.70i)T + (-1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (3.76 - 1.00i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.56 + 2.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.46 + 0.684i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (6.30 + 0.551i)T + (16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (0.0168 + 0.0240i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.496 - 0.416i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (6.55 + 3.78i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.88 - 4.88i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.29 - 6.29i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.26 - 2.28i)T + (14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (1.16 + 13.3i)T + (-46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-0.241 + 0.168i)T + (18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (5.17 - 4.34i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0829 - 0.470i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.83 - 0.685i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (5.21 - 0.918i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.79 + 2.70i)T + (46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (11.2 + 4.10i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.42 - 12.7i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (3.79 - 1.38i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.733 - 8.38i)T + (-95.5 - 16.8i)T^{2} \) |
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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
−9.633492325398058555014582258946, −9.067860356386969664970058760129, −8.479069802960618282208514788863, −7.18502780839100645937241623571, −6.48735858181738828423313134933, −5.51703316966576074308595083663, −4.58395022143734617523468899797, −3.50829560195951417230651063384, −2.75539984462401996666393337186, −0.05738705360603606261527120377,
1.71675518680116543360332878288, 2.61176820331455800843576488424, 3.65466343840630252887074912375, 4.75342108597851428125875096551, 6.10628469421037208064816230080, 7.12392179814583942076922188796, 7.47880644759350681550031694156, 8.724108640195244631790106773156, 9.415094799763359733096076248132, 10.23337889641975876142748323063
