| L(s) = 1 | + (0.321 − 1.82i)2-s + (2.27 + 1.59i)3-s + (−1.33 − 0.485i)4-s + (3.63 − 3.63i)6-s + (0.309 − 3.54i)7-s + (0.535 − 0.927i)8-s + (1.61 + 4.44i)9-s + (4.83 + 2.79i)11-s + (−2.26 − 3.23i)12-s + (−2.98 − 1.08i)13-s + (−6.35 − 1.70i)14-s + (−3.69 − 3.10i)16-s + (1.03 + 2.83i)17-s + (8.62 − 1.52i)18-s + (0.0922 − 0.131i)19-s + ⋯ |
| L(s) = 1 | + (0.227 − 1.28i)2-s + (1.31 + 0.920i)3-s + (−0.667 − 0.242i)4-s + (1.48 − 1.48i)6-s + (0.117 − 1.33i)7-s + (0.189 − 0.327i)8-s + (0.539 + 1.48i)9-s + (1.45 + 0.841i)11-s + (−0.654 − 0.934i)12-s + (−0.826 − 0.300i)13-s + (−1.69 − 0.454i)14-s + (−0.923 − 0.775i)16-s + (0.250 + 0.687i)17-s + (2.03 − 0.358i)18-s + (0.0211 − 0.0302i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42484 - 1.79177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42484 - 1.79177i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 37 | \( 1 + (-1.74 - 5.82i)T \) |
| good | 2 | \( 1 + (-0.321 + 1.82i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-2.27 - 1.59i)T + (1.02 + 2.81i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 3.54i)T + (-6.89 - 1.21i)T^{2} \) |
| 11 | \( 1 + (-4.83 - 2.79i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.98 + 1.08i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.03 - 2.83i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.0922 + 0.131i)T + (-6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (3.22 + 5.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.02 - 7.54i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (3.60 + 3.60i)T + 31iT^{2} \) |
| 41 | \( 1 + (-0.214 + 0.589i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 - 0.198T + 43T^{2} \) |
| 47 | \( 1 + (-0.516 - 0.138i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.212 - 2.42i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (0.179 + 2.05i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (0.187 + 0.402i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-14.0 - 1.22i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.219 - 1.24i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (6.87 - 6.87i)T - 73iT^{2} \) |
| 79 | \( 1 + (6.64 + 0.581i)T + (77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (8.81 + 4.10i)T + (53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (5.12 - 0.448i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (6.38 - 3.68i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.06832309858248068245613222081, −9.461456591864001353395584236867, −8.537564214034163943563891180319, −7.49739646852980062059059385325, −6.73537628611169447756599294305, −4.71288005477762439543027161842, −4.11919472087178729052409131765, −3.59847183997583098846179751084, −2.51164556055765192931866269267, −1.38966285501135517765811641978,
1.75940930633555342653149518727, 2.70361686571543391646914128842, 3.95614661885224646747459835597, 5.37689301882945188155105838861, 6.13118725944491447498417545630, 6.94956613785394023065914656425, 7.67024036659984316150450204802, 8.394795758720229258858214402252, 9.045767222077290372314182430869, 9.570438433120014382676803572648
