L(s) = 1 | − 1.08i·3-s + 1.08i·5-s − 7-s + 1.82·9-s + 5.22i·11-s − 6.30i·13-s + 1.17·15-s + 3.65·17-s + 4.14i·19-s + 1.08i·21-s − 1.17·23-s + 3.82·25-s − 5.22i·27-s + 8.28i·29-s + 5.65·31-s + ⋯ |
L(s) = 1 | − 0.624i·3-s + 0.484i·5-s − 0.377·7-s + 0.609·9-s + 1.57i·11-s − 1.74i·13-s + 0.302·15-s + 0.886·17-s + 0.950i·19-s + 0.236i·21-s − 0.244·23-s + 0.765·25-s − 1.00i·27-s + 1.53i·29-s + 1.01·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63752\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 1.08iT - 3T^{2} \) |
| 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 11 | \( 1 - 5.22iT - 11T^{2} \) |
| 13 | \( 1 + 6.30iT - 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 4.14iT - 19T^{2} \) |
| 23 | \( 1 + 1.17T + 23T^{2} \) |
| 29 | \( 1 - 8.28iT - 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 2.16iT - 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 5.22iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 4.32iT - 53T^{2} \) |
| 59 | \( 1 - 6.30iT - 59T^{2} \) |
| 61 | \( 1 + 7.20iT - 61T^{2} \) |
| 67 | \( 1 + 7.39iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.41iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.32876765713086661865512205290, −9.472070271214050112182526675492, −8.140338995648039637841101440132, −7.48640582211781263320515150529, −6.88485917240661119438985936688, −5.88253989446817293065956727907, −4.86674656817961731328386399614, −3.61024147735389999189844250735, −2.54655482498280317382189571805, −1.20216301084953963803582209224,
1.00603784400427532290366559574, 2.73586866346125114337555037798, 3.96903067483209967287134520277, 4.58121578400067069975973221691, 5.76694528978135966106523776545, 6.57541740603830524177403390152, 7.61694612204372121044984870151, 8.731372102233984277719821839871, 9.246486752866700899245840494681, 9.989140511287009244354355653196