| L(s) = 1 | − i·2-s − 4-s + (−2.17 − 0.527i)5-s + 4.99i·7-s + i·8-s + (−0.527 + 2.17i)10-s − 2.41·11-s + 3.44i·13-s + 4.99·14-s + 16-s − 1.84i·17-s − 0.288·19-s + (2.17 + 0.527i)20-s + 2.41i·22-s − 0.170i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.971 − 0.235i)5-s + 1.88i·7-s + 0.353i·8-s + (−0.166 + 0.687i)10-s − 0.726·11-s + 0.955i·13-s + 1.33·14-s + 0.250·16-s − 0.448i·17-s − 0.0662·19-s + (0.485 + 0.117i)20-s + 0.513i·22-s − 0.0355i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2166147942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2166147942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (2.17 + 0.527i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 - 4.99iT - 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 - 3.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.84iT - 17T^{2} \) |
| 19 | \( 1 + 0.288T + 19T^{2} \) |
| 23 | \( 1 + 0.170iT - 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 8.57iT - 43T^{2} \) |
| 47 | \( 1 - 1.88iT - 47T^{2} \) |
| 53 | \( 1 - 3.73iT - 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 9.08iT - 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 + 13.2iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 3.21iT - 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 - 7.95iT - 97T^{2} \) |
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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
−9.033264751608359239204815672489, −8.330086620227899306065716438672, −7.82404354681007331194214747935, −6.65841373944389168482791185344, −5.85374935263228634634986340822, −4.85281631626669268198458720827, −4.52259728500057689944315948623, −3.12936454276188541009872460423, −2.68914020420190460185185733047, −1.56927542164419525541655200765,
0.079072587717713775034357627849, 1.03407233113350389293143343482, 2.93328358554996553749701656782, 3.73436271309742886607389235836, 4.40385802444672830589372150813, 5.12077588073187158594306173746, 6.22491279702337189157271704684, 7.04725877130103298254074881874, 7.45493712524958163750775710991, 8.133525448020661108176784440075
