L(s) = 1 | + 2.52·5-s + (1.07 + 2.41i)7-s − 5.71·11-s + (−2.45 − 4.24i)13-s + (−2.49 − 4.32i)17-s + (0.00383 − 0.00664i)19-s + 0.667·23-s + 1.36·25-s + (−3.85 + 6.66i)29-s + (−3.88 + 6.72i)31-s + (2.71 + 6.09i)35-s + (−3.19 + 5.53i)37-s + (−5.21 − 9.02i)41-s + (−4.42 + 7.67i)43-s + (−1.08 − 1.87i)47-s + ⋯ |
L(s) = 1 | + 1.12·5-s + (0.407 + 0.913i)7-s − 1.72·11-s + (−0.680 − 1.17i)13-s + (−0.605 − 1.04i)17-s + (0.000880 − 0.00152i)19-s + 0.139·23-s + 0.273·25-s + (−0.715 + 1.23i)29-s + (−0.697 + 1.20i)31-s + (0.459 + 1.03i)35-s + (−0.525 + 0.909i)37-s + (−0.813 − 1.40i)41-s + (−0.675 + 1.16i)43-s + (−0.157 − 0.272i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3727034510\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3727034510\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.07 - 2.41i)T \) |
good | 5 | \( 1 - 2.52T + 5T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 + (2.45 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.49 + 4.32i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.00383 + 0.00664i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.667T + 23T^{2} \) |
| 29 | \( 1 + (3.85 - 6.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.88 - 6.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.19 - 5.53i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.21 + 9.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.42 - 7.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.08 + 1.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.69 - 6.40i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.261 + 0.453i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.49 - 7.78i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.54 - 4.41i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + (1.52 + 2.63i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.08 - 5.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.448i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.19 - 2.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.32 + 7.49i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.000202843406546584095753101123, −8.446314101892346780128248156116, −7.55094145988909832840633435360, −6.88880514895215983117084813500, −5.66899771931185472103656134224, −5.31067055781769366501578728092, −4.88946887788108900970853799161, −3.07934260486041685264114969262, −2.60179683315015298342951682521, −1.68675016062024851816101863880,
0.099064403954509078023102568877, 1.91841542299580093057106919183, 2.20724878925741295177689809362, 3.67356523827578130046656293156, 4.56502848408791621635286065744, 5.29202461909312724589797346194, 6.05936924037570488796730727005, 6.88894953607209350992820366301, 7.64536751368051067464460877060, 8.272435953616572648955172085799