L(s) = 1 | + (1 − 1.41i)3-s + (−2.41 + 2.41i)5-s + 7-s + (−1.00 − 2.82i)9-s + (−1.82 − 1.82i)11-s + (−1.58 + 1.58i)13-s + (1 + 5.82i)15-s − 6.82i·17-s + (2.41 + 2.41i)19-s + (1 − 1.41i)21-s + 3.65i·23-s − 6.65i·25-s + (−5.00 − 1.41i)27-s + (−3 − 3i)29-s − 10.4i·31-s + ⋯ |
L(s) = 1 | + (0.577 − 0.816i)3-s + (−1.07 + 1.07i)5-s + 0.377·7-s + (−0.333 − 0.942i)9-s + (−0.551 − 0.551i)11-s + (−0.439 + 0.439i)13-s + (0.258 + 1.50i)15-s − 1.65i·17-s + (0.553 + 0.553i)19-s + (0.218 − 0.308i)21-s + 0.762i·23-s − 1.33i·25-s + (−0.962 − 0.272i)27-s + (−0.557 − 0.557i)29-s − 1.88i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7437242271\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7437242271\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 1.41i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (2.41 - 2.41i)T - 5iT^{2} \) |
| 11 | \( 1 + (1.82 + 1.82i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.58 - 1.58i)T - 13iT^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 + (-2.41 - 2.41i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.65iT - 23T^{2} \) |
| 29 | \( 1 + (3 + 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (3.82 + 3.82i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + (1.82 - 1.82i)T - 43iT^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + (0.171 - 0.171i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.07 + 4.07i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.414 - 0.414i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (-10.8 + 10.8i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.230339890826066631599940103774, −8.125481244676711164206579142674, −7.58863759015006663191352978647, −7.17772531943815417071473098653, −6.20295677965717549238787329398, −5.06517143850924504364643675346, −3.72636817996185279744515505712, −3.08062174528199561501755240421, −2.08040230191164102626579712332, −0.27381194923424441313464692011,
1.65867709854020004461908263495, 3.14075889838702447559997436709, 3.98119724810994481960812186008, 4.95860008739282610995746630427, 5.17587520473510499240657918423, 6.89332091096994920880624372671, 7.88731122886830277294680517745, 8.410100445067784935103036396420, 8.858948613893322074536987979384, 10.00839660866598322628309589549