L(s) = 1 | + i·5-s − 6i·13-s + 2i·19-s + 6·23-s − 25-s + 6i·29-s − 6i·37-s − 6·41-s − 8i·43-s + 6·47-s − 7·49-s − 6i·53-s − 12i·59-s − 12i·61-s + 6·65-s + ⋯ |
L(s) = 1 | + 0.447i·5-s − 1.66i·13-s + 0.458i·19-s + 1.25·23-s − 0.200·25-s + 1.11i·29-s − 0.986i·37-s − 0.937·41-s − 1.21i·43-s + 0.875·47-s − 49-s − 0.824i·53-s − 1.56i·59-s − 1.53i·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.651159801\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651159801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−8.573091330886350399659647027354, −7.945330926207440916273343694046, −7.16258733738589617729740237835, −6.48922672055492464537477326866, −5.45350893226106825946899862817, −5.03821679825575861947731981153, −3.62777866060166136172783597331, −3.15150350402670343791280774122, −2.00814878055854722772814245621, −0.60935198174608818132866900175,
1.07744917579644821256237177127, 2.15290710932403608688646194148, 3.21835517779007736385053037554, 4.38637631019046432340169399509, 4.75029737616232500864283352058, 5.86713601413244357990765643181, 6.63781115949250136543882480737, 7.28739593104935365220407784665, 8.201738405221438610130321612715, 8.995023667265877932221464534425