| L(s) = 1 | + i·3-s + (−2 − i)5-s − 2i·7-s − 9-s − 4·11-s + (1 − 2i)15-s + 2i·17-s + 4·19-s + 2·21-s + i·23-s + (3 + 4i)25-s − i·27-s − 6·29-s − 4·31-s − 4i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + (0.258 − 0.516i)15-s + 0.485i·17-s + 0.917·19-s + 0.436·21-s + 0.208i·23-s + (0.600 + 0.800i)25-s − 0.192i·27-s − 1.11·29-s − 0.718·31-s − 0.696i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9665471980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9665471980\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.951517201051043144691401280285, −8.083253211482630835481786128587, −7.58842223226212585155607326141, −6.89621811238670327794173401304, −5.44261141396577850219621026985, −5.20824547283843736992777904900, −3.92276071525409519060252635695, −3.72024204209046853019200863999, −2.41484005997367363709808728167, −0.838323004738218513609693579865,
0.42267850469693064444461785914, 2.07105756158970407364213909488, 2.90423488835218237385501728560, 3.67206706567516067521709200961, 4.99536815842966529969700522713, 5.49586936737580319881668136875, 6.55047349675073203641417324449, 7.28142041164241440176422774600, 7.87812857723712230194175876704, 8.453474631533029587853277338188
