L(s) = 1 | + 8.86·3-s + 7.47i·5-s + 51.5·9-s − 61.2i·11-s + 12.0i·13-s + 66.2i·15-s + 100. i·17-s + 73.6·19-s + 96.8i·23-s + 69.1·25-s + 217.·27-s + 284.·29-s − 256.·31-s − 542. i·33-s + 408.·37-s + ⋯ |
L(s) = 1 | + 1.70·3-s + 0.668i·5-s + 1.90·9-s − 1.67i·11-s + 0.257i·13-s + 1.14i·15-s + 1.43i·17-s + 0.889·19-s + 0.877i·23-s + 0.552·25-s + 1.54·27-s + 1.82·29-s − 1.48·31-s − 2.86i·33-s + 1.81·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.184306324\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.184306324\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 8.86T + 27T^{2} \) |
| 5 | \( 1 - 7.47iT - 125T^{2} \) |
| 11 | \( 1 + 61.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 12.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 100. iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 73.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96.8iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 284.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 256.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 120. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 308. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 26.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 311.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 444.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 488. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 585. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 416. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 819. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 318. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 218.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 19.4iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 306. iT - 9.12e5T^{2} \) |
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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
−9.804173818428458809380224666086, −8.900976750631652161446585042437, −8.355950694281190972444169128140, −7.60706061278328061572131759429, −6.65122825404529539781212355184, −5.61062137439082604483092864369, −3.98603391234728517934569371317, −3.32990985825143778462761108178, −2.56972563662762986418941434081, −1.24053866681453803976166166449,
1.08014124294446454027031168066, 2.32626250644420088550406414735, 3.08596669164628790752648265881, 4.44143869616635818058039382824, 4.93453744903108940943715231221, 6.71283726817417102567273077285, 7.54435335546274405456446297234, 8.114664951784288398157967332792, 9.190350675635072180695876563803, 9.485533414722842687997460018680