| L(s) = 1 | + 1.73i·3-s + 1.73i·5-s + 6·9-s − 15·11-s − 13.8i·13-s − 2.99·15-s − 29.4i·17-s − 15.5i·19-s + 9·23-s + 22·25-s + 25.9i·27-s − 6·29-s − 12.1i·31-s − 25.9i·33-s + 31·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.346i·5-s + 0.666·9-s − 1.36·11-s − 1.06i·13-s − 0.199·15-s − 1.73i·17-s − 0.820i·19-s + 0.391·23-s + 0.880·25-s + 0.962i·27-s − 0.206·29-s − 0.391i·31-s − 0.787i·33-s + 0.837·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.533020812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533020812\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73iT - 9T^{2} \) |
| 5 | \( 1 - 1.73iT - 25T^{2} \) |
| 11 | \( 1 + 15T + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 29.4iT - 289T^{2} \) |
| 19 | \( 1 + 15.5iT - 361T^{2} \) |
| 23 | \( 1 - 9T + 529T^{2} \) |
| 29 | \( 1 + 6T + 841T^{2} \) |
| 31 | \( 1 + 12.1iT - 961T^{2} \) |
| 37 | \( 1 - 31T + 1.36e3T^{2} \) |
| 41 | \( 1 - 55.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 57T + 2.80e3T^{2} \) |
| 59 | \( 1 + 81.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 81.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 49T + 4.48e3T^{2} \) |
| 71 | \( 1 - 126T + 5.04e3T^{2} \) |
| 73 | \( 1 - 25.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 73T + 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 57.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 27.7iT - 9.40e3T^{2} \) |
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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
−9.896969320399893350720671890267, −9.474486033397285778157713980754, −8.192426573613126138188888441368, −7.45310357021968186683501839608, −6.61207532360505682354586947870, −5.13463604168609752515640732691, −4.87600901878805658871182527061, −3.31947079864042299535809992419, −2.55794864191292472332526844256, −0.56609539215108918849842125416,
1.27089078347512330083847822583, 2.29324926893561150076887089496, 3.81346157555010461505081736731, 4.77252609846375273769618876641, 5.86739019661196875424114325445, 6.75386661697508725834906265798, 7.67691664093910484777124264400, 8.333091457870875202040225664075, 9.289471349813634782304310976298, 10.33675957284372467901257231661
