L(s) = 1 | − 0.892·2-s + (0.497 + 1.65i)3-s − 1.20·4-s + i·5-s + (−0.444 − 1.48i)6-s + i·7-s + 2.85·8-s + (−2.50 + 1.65i)9-s − 0.892i·10-s + (0.586 − 3.26i)11-s + (−0.598 − 1.99i)12-s − 5.84i·13-s − 0.892i·14-s + (−1.65 + 0.497i)15-s − 0.147·16-s + 4.41·17-s + ⋯ |
L(s) = 1 | − 0.631·2-s + (0.287 + 0.957i)3-s − 0.601·4-s + 0.447i·5-s + (−0.181 − 0.604i)6-s + 0.377i·7-s + 1.01·8-s + (−0.834 + 0.550i)9-s − 0.282i·10-s + (0.176 − 0.984i)11-s + (−0.172 − 0.576i)12-s − 1.62i·13-s − 0.238i·14-s + (−0.428 + 0.128i)15-s − 0.0367·16-s + 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004242323\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004242323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.497 - 1.65i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.586 + 3.26i)T \) |
good | 2 | \( 1 + 0.892T + 2T^{2} \) |
| 13 | \( 1 + 5.84iT - 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 1.11iT - 19T^{2} \) |
| 23 | \( 1 + 7.41iT - 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 + 0.0544T + 31T^{2} \) |
| 37 | \( 1 + 4.09T + 37T^{2} \) |
| 41 | \( 1 - 0.969T + 41T^{2} \) |
| 43 | \( 1 - 0.978iT - 43T^{2} \) |
| 47 | \( 1 - 11.0iT - 47T^{2} \) |
| 53 | \( 1 + 0.255iT - 53T^{2} \) |
| 59 | \( 1 + 1.86iT - 59T^{2} \) |
| 61 | \( 1 + 5.07iT - 61T^{2} \) |
| 67 | \( 1 - 5.79T + 67T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 + 7.44iT - 73T^{2} \) |
| 79 | \( 1 - 7.49iT - 79T^{2} \) |
| 83 | \( 1 - 5.81T + 83T^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 1.50T + 97T^{2} \) |
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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
−9.800713330179497217589705656275, −9.022229641643991998794101893035, −8.173697364662440460134892556457, −7.966713035816800541067896388382, −6.35810259225953061625124450450, −5.42677029557273123140184354157, −4.66763370627989862125802868459, −3.45622549861992128131046766929, −2.78115886274599132537485060672, −0.66915995403825363012877031328,
1.11293706903558387545790610000, 1.90685925535813535021747849488, 3.61768579905475293447787797644, 4.54628591455784033401802254250, 5.56426638436293548693213885829, 6.87123084675015615493615517377, 7.34349659606425268745750199302, 8.233748220925614632886739351596, 8.888250814904363622120805611395, 9.647656867936686276796083972937