L(s) = 1 | + 1.41i·2-s + (1.41 + i)3-s − 2.00·4-s − 1.41i·5-s + (−1.41 + 2.00i)6-s − 2i·7-s − 2.82i·8-s + (1.00 + 2.82i)9-s + 2.00·10-s + 5.65·11-s + (−2.82 − 2.00i)12-s + 2·13-s + 2.82·14-s + (1.41 − 2.00i)15-s + 4.00·16-s − 5.65i·17-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (0.816 + 0.577i)3-s − 1.00·4-s − 0.632i·5-s + (−0.577 + 0.816i)6-s − 0.755i·7-s − 1.00i·8-s + (0.333 + 0.942i)9-s + 0.632·10-s + 1.70·11-s + (−0.816 − 0.577i)12-s + 0.554·13-s + 0.755·14-s + (0.365 − 0.516i)15-s + 1.00·16-s − 1.37i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78986 + 0.926504i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78986 + 0.926504i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 67 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.65iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.961283666081856270902764827382, −9.165074091391689096149182110559, −8.895807135471972084605273114470, −7.81941394228771602618647940015, −7.06940000786716704339351723361, −6.12042785738433281686824664655, −4.73589338871446679481855148646, −4.31094421622386009377705044832, −3.27396045296346793535723177972, −1.15243595717591120963763979206,
1.45990207639139647581672111987, 2.31273361621025530140938294406, 3.60731602762211980599705831166, 4.03903389093982803431406964090, 5.95736697000494878546597121092, 6.48074209884614893404768362839, 7.983898668334688126977541636950, 8.490312850385421187270544960249, 9.365454298812358188425009868430, 10.01364198579013433686129818675