| L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s + 4i·11-s + 12-s + i·15-s + 16-s + 6·17-s − i·18-s − 4i·19-s + i·20-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.20i·11-s + 0.288·12-s + 0.258i·15-s + 0.250·16-s + 1.45·17-s − 0.235i·18-s − 0.917i·19-s + 0.223i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9824146383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9824146383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.322744553475654579019178608893, −7.63143783172067378376054679059, −6.89345169608282922067128485984, −5.97208297641197493495612767500, −5.21464337660547116155464114702, −4.60823928175496984421791453168, −3.90967077195863446539368182483, −2.86520048737423861042903124174, −1.85225113094165631182200785856, −0.985622654308429601258639247182,
0.34515317747471623257156338591, 1.56538039233603013895554301316, 3.01022719892157244847149251919, 3.73093990139767201189774484145, 4.56725924546548230116991366418, 5.67281168118077168719861733110, 5.90255786825983424374140417080, 6.53871494493932803496032202455, 7.50233812875175255108011869558, 8.072109998916235197105837615063
