L(s) = 1 | − 1.09·3-s + (2.20 + 0.370i)5-s − i·7-s − 1.80·9-s − 1.07i·11-s − 3.77·13-s + (−2.41 − 0.405i)15-s + 5.08i·17-s − 0.279i·19-s + 1.09i·21-s − 4.13i·23-s + (4.72 + 1.63i)25-s + 5.25·27-s + 4.87i·29-s + 10.1·31-s + ⋯ |
L(s) = 1 | − 0.631·3-s + (0.986 + 0.165i)5-s − 0.377i·7-s − 0.601·9-s − 0.323i·11-s − 1.04·13-s + (−0.622 − 0.104i)15-s + 1.23i·17-s − 0.0641i·19-s + 0.238i·21-s − 0.861i·23-s + (0.944 + 0.327i)25-s + 1.01·27-s + 0.904i·29-s + 1.81·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483327583\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483327583\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.20 - 0.370i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 11 | \( 1 + 1.07iT - 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 - 5.08iT - 17T^{2} \) |
| 19 | \( 1 + 0.279iT - 19T^{2} \) |
| 23 | \( 1 + 4.13iT - 23T^{2} \) |
| 29 | \( 1 - 4.87iT - 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 + 4.78iT - 47T^{2} \) |
| 53 | \( 1 - 5.30T + 53T^{2} \) |
| 59 | \( 1 + 5.22iT - 59T^{2} \) |
| 61 | \( 1 - 14.5iT - 61T^{2} \) |
| 67 | \( 1 + 9.74T + 67T^{2} \) |
| 71 | \( 1 - 4.18T + 71T^{2} \) |
| 73 | \( 1 + 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 + 2.13T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 9.92iT - 97T^{2} \) |
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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
−9.029417353708086251579438234967, −8.385899621743641493210059167971, −7.39036773182469637981893654367, −6.42966085656955121291545692911, −6.04817852360687711224569421363, −5.16019770480096696131857317027, −4.41269660668529878588380751008, −3.07355671763779868813656017886, −2.22891110639500553119301704066, −0.841589972422607768966875798121,
0.78200997004611337814494732473, 2.30202337011635077070698162093, 2.87498052809874807373109546986, 4.53356666289476524544148661765, 5.13065318411284001172922997832, 5.84760677983812601821289086533, 6.48593169341653499199391170272, 7.41638711448599293410293077737, 8.262235669471139497801962713194, 9.361031515915007978899888646763