| L(s) = 1 | − i·3-s + (1 − 2i)5-s − i·7-s − 9-s + 6·11-s − 2i·13-s + (−2 − i)15-s − 4i·17-s − 6·19-s − 21-s + (−3 − 4i)25-s + i·27-s + 2·29-s + 10·31-s − 6i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.447 − 0.894i)5-s − 0.377i·7-s − 0.333·9-s + 1.80·11-s − 0.554i·13-s + (−0.516 − 0.258i)15-s − 0.970i·17-s − 1.37·19-s − 0.218·21-s + (−0.600 − 0.800i)25-s + 0.192i·27-s + 0.371·29-s + 1.79·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.883248210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883248210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 16iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.037316624036424670548798856841, −8.391286020574861295884542478247, −7.55394851584204516514435058663, −6.40801056846433653809037194446, −6.22190877898798408188190991752, −4.83581668636817029977509134717, −4.25095807035757933035455119348, −2.93737312533058031418294316443, −1.65001263413998875062560435512, −0.75683000651717450731875337317,
1.63339636286715631086532618820, 2.68373142689175862657521282022, 3.86335002827774524147219868801, 4.37051167475054357166874939529, 5.77297122144178884552119185260, 6.41465917000920879765685209947, 6.89254801942769031751785061203, 8.275276814594937286758130429113, 8.901764698100648976381902966375, 9.630484987504833201068314081556
