| L(s) = 1 | + 2.82i·5-s + 2.64·7-s − 11.2i·11-s + 14·13-s − 5.65i·17-s + 21.1·19-s − 11.2i·23-s + 17·25-s + 21.2i·29-s − 31.7·31-s + 7.48i·35-s − 60·37-s − 8.48i·41-s − 21.1·43-s − 52.3i·47-s + ⋯ |
| L(s) = 1 | + 0.565i·5-s + 0.377·7-s − 1.02i·11-s + 1.07·13-s − 0.332i·17-s + 1.11·19-s − 0.488i·23-s + 0.680·25-s + 0.731i·29-s − 1.02·31-s + 0.213i·35-s − 1.62·37-s − 0.206i·41-s − 0.492·43-s − 1.11i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.213485146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213485146\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 2.82iT - 25T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 14T + 169T^{2} \) |
| 17 | \( 1 + 5.65iT - 289T^{2} \) |
| 19 | \( 1 - 21.1T + 361T^{2} \) |
| 23 | \( 1 + 11.2iT - 529T^{2} \) |
| 29 | \( 1 - 21.2iT - 841T^{2} \) |
| 31 | \( 1 + 31.7T + 961T^{2} \) |
| 37 | \( 1 + 60T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.48iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 67.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 10T + 3.72e3T^{2} \) |
| 67 | \( 1 - 68.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 86.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42T + 5.32e3T^{2} \) |
| 79 | \( 1 - 89.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 89.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 126T + 9.40e3T^{2} \) |
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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
−8.673946947527210654029315665431, −8.334272056464079797200812797437, −7.17675893091894633554599440768, −6.66550249117339503866336719232, −5.62220650021584548563078901357, −5.04203533927824156488103526507, −3.61784702470571300595214299146, −3.22213322285868767786353328516, −1.85796536741045534152926233297, −0.65872557759073809367234778678,
1.06199693249631494504189405223, 1.89752806920385654684836710101, 3.27161248451363389328863489820, 4.17990122999406174506943009693, 5.04372008737311590851597259409, 5.72384130122422520455390727884, 6.76494734580966860465049452853, 7.54770359066098188160430700779, 8.274189329847485194175450237144, 9.080583049131088013092959654874
