| L(s) = 1 | + 2.82·5-s − i·7-s − 1.41i·11-s + 2i·13-s + 2.82i·17-s + 4·19-s + 1.41·23-s + 3.00·25-s + 1.41·29-s − 2.82i·35-s + 10i·37-s + 5.65i·41-s + 2·43-s + 2.82·47-s − 49-s + ⋯ |
| L(s) = 1 | + 1.26·5-s − 0.377i·7-s − 0.426i·11-s + 0.554i·13-s + 0.685i·17-s + 0.917·19-s + 0.294·23-s + 0.600·25-s + 0.262·29-s − 0.478i·35-s + 1.64i·37-s + 0.883i·41-s + 0.304·43-s + 0.412·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.591743614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.591743614\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 11.3iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.435493205391657510535863313537, −7.83413177308856446359254445266, −6.71237377097565645252522612765, −6.38832340692914058804913334618, −5.48407526939148692751660258591, −4.86580638978457067379345072245, −3.81245103387978208764772076043, −2.92134837815372225782292882102, −1.91719057252140360797009512174, −1.04827222856337348425303151538,
0.885933188659881201346982323460, 2.08575011881872810591284746599, 2.69456685688658018429377411471, 3.75468363056921986166153969760, 4.91761113335368298621499944675, 5.51119939543041355124400338044, 6.01896351764703913675286487217, 7.02328195974621541089942314225, 7.54469716256462548443851557374, 8.582057062241957721468718889668
