| L(s) = 1 | + 2.64i·7-s + 3·9-s + 4·11-s − 5.29i·23-s − 5·25-s + 10.5i·29-s − 10.5i·37-s − 12·43-s − 7.00·49-s − 10.5i·53-s + 7.93i·63-s − 4·67-s + 5.29i·71-s + 10.5i·77-s − 15.8i·79-s + ⋯ |
| L(s) = 1 | + 0.999i·7-s + 9-s + 1.20·11-s − 1.10i·23-s − 25-s + 1.96i·29-s − 1.73i·37-s − 1.82·43-s − 49-s − 1.45i·53-s + 0.999i·63-s − 0.488·67-s + 0.627i·71-s + 1.20i·77-s − 1.78i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30065 + 0.237597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30065 + 0.237597i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 5.29iT - 23T^{2} \) |
| 29 | \( 1 - 10.5iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10.5iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.5iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.32180775967608868365646478578, −11.50064929263654063357137787404, −10.31051710828699044062156525461, −9.312480829385573256702764495218, −8.546395830076278245343632819159, −7.16150935108467611770633917749, −6.23831579269273872917902894561, −4.92680883399711132158191577086, −3.62797910414392303563635931634, −1.83878120835424374972243439361,
1.45605785642116846050891671960, 3.67547683248527082149287337927, 4.53270465427190292181413969951, 6.21582404203390083723612488505, 7.15168892393948979255100184924, 8.083173450993270659605003738941, 9.582358494613024752970500930116, 10.04871488055362295682351441334, 11.34436108247883167850609453958, 12.06557483860527034299620531922
