| L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.266 + 1.50i)3-s + (−0.939 − 0.342i)4-s + (−1.79 + 1.50i)5-s − 1.53·6-s + (1.93 − 1.62i)7-s + (0.5 − 0.866i)8-s + (0.613 − 0.223i)9-s + (−1.17 − 2.03i)10-s + (−0.560 + 0.970i)11-s + (0.266 − 1.50i)12-s + (1.70 + 0.620i)13-s + (1.26 + 2.19i)14-s + (−2.75 − 2.31i)15-s + (0.766 + 0.642i)16-s + (2.81 − 1.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.153 + 0.871i)3-s + (−0.469 − 0.171i)4-s + (−0.804 + 0.674i)5-s − 0.625·6-s + (0.733 − 0.615i)7-s + (0.176 − 0.306i)8-s + (0.204 − 0.0744i)9-s + (−0.371 − 0.642i)10-s + (−0.168 + 0.292i)11-s + (0.0768 − 0.435i)12-s + (0.473 + 0.172i)13-s + (0.338 + 0.586i)14-s + (−0.711 − 0.596i)15-s + (0.191 + 0.160i)16-s + (0.683 − 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.577433 + 0.634087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577433 + 0.634087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (-2.33 + 5.61i)T \) |
| good | 3 | \( 1 + (-0.266 - 1.50i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.79 - 1.50i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 1.62i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.560 - 0.970i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 0.620i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.81 + 1.02i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (0.971 + 5.51i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (4.55 + 7.88i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.52 - 7.83i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.53T + 31T^{2} \) |
| 41 | \( 1 + (-6.98 - 2.54i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + (-0.194 - 0.337i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.23 + 7.74i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.41 - 5.38i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.45 - 0.892i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.62 + 4.72i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.448 + 2.54i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + 0.709T + 73T^{2} \) |
| 79 | \( 1 + (-3.39 + 2.84i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.81 + 2.47i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (9.95 + 8.35i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.34 - 12.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.79126996075159091001289688913, −14.44345191612597347524410721406, −12.85159607982860053185772858297, −11.20092133269274967520891102133, −10.45818334099382687659737312304, −9.136971278839220371114429063825, −7.81507301990169446241714833891, −6.84975231250723620386439949545, −4.85644310162187915331938115454, −3.76372049722118746170289440162,
1.67447562853816073528053822664, 3.93218650718567271627324299217, 5.66185389277856697238812700976, 7.898322386672053572906011818409, 8.174048153995259133981210448840, 9.806356439373998621442636560577, 11.37582881874450010097122845910, 12.10383584860166847713992591356, 12.93741711676873873325698142158, 13.98948526492443893576606521592
