L(s) = 1 | + (−1.33 + 2.31i)5-s + (2.54 − 0.714i)7-s + (−1.99 − 3.45i)11-s + (1.00 + 1.73i)13-s + (3.57 − 6.18i)17-s + (−4.01 − 6.96i)19-s + (−0.443 + 0.768i)23-s + (−1.06 − 1.83i)25-s + (1.35 − 2.33i)29-s − 1.22·31-s + (−1.74 + 6.84i)35-s + (5.26 + 9.11i)37-s + (1.43 + 2.48i)41-s + (3.40 − 5.88i)43-s + 12.1·47-s + ⋯ |
L(s) = 1 | + (−0.596 + 1.03i)5-s + (0.962 − 0.270i)7-s + (−0.600 − 1.04i)11-s + (0.277 + 0.480i)13-s + (0.866 − 1.50i)17-s + (−0.922 − 1.59i)19-s + (−0.0925 + 0.160i)23-s + (−0.212 − 0.367i)25-s + (0.250 − 0.434i)29-s − 0.220·31-s + (−0.295 + 1.15i)35-s + (0.865 + 1.49i)37-s + (0.224 + 0.388i)41-s + (0.518 − 0.898i)43-s + 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.528012764\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.528012764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.54 + 0.714i)T \) |
good | 5 | \( 1 + (1.33 - 2.31i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.00 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.57 + 6.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.01 + 6.96i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.443 - 0.768i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.35 + 2.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + (-5.26 - 9.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.43 - 2.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.40 + 5.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + (-2.38 + 4.13i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 9.58T + 59T^{2} \) |
| 61 | \( 1 + 9.49T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 4.62T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 + 3.48i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 + (-5.26 + 9.12i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.72 + 2.99i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.12 - 1.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.324491891497738193312734477229, −8.495037827988803920064278105147, −7.68120874304972101439627280852, −7.14828483315379035173230275055, −6.23997900449097665500021111785, −5.15856023124373998955273854826, −4.34563049479968890536973782629, −3.21343216260525810852539732798, −2.46736050692940183719193862595, −0.69667357538962159742922146170,
1.21431811404907990000608027953, 2.22124535468393056404146505681, 3.88972110139364383333803955105, 4.40824475015706528522052642442, 5.42405675907082316106609161905, 6.02518296135668383174773728757, 7.56779466201890480477578417118, 7.988947445500987803395586582678, 8.520740212640226232264084572001, 9.447469352727223832281726877465