L(s) = 1 | + (−0.707 − 0.707i)3-s − 1.99i·9-s + 1.73·11-s + (2.82 − 2.82i)13-s + (−1.22 + 1.22i)17-s − 5.19i·19-s + (−4.89 + 4.89i)23-s + (−3.53 + 3.53i)27-s + 6.92·29-s + 4i·31-s + (−1.22 − 1.22i)33-s + (−5.65 − 5.65i)37-s − 4.00·39-s + 3·41-s + (−5.65 − 5.65i)43-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s − 0.666i·9-s + 0.522·11-s + (0.784 − 0.784i)13-s + (−0.297 + 0.297i)17-s − 1.19i·19-s + (−1.02 + 1.02i)23-s + (−0.680 + 0.680i)27-s + 1.28·29-s + 0.718i·31-s + (−0.213 − 0.213i)33-s + (−0.929 − 0.929i)37-s − 0.640·39-s + 0.468·41-s + (−0.862 − 0.862i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219887309\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219887309\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + (-2.82 + 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.22 - 1.22i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 + (4.89 - 4.89i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (5.65 + 5.65i)T + 37iT^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + (5.65 + 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.89 - 4.89i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + (4.94 - 4.94i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (6.12 + 6.12i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (10.6 + 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 3iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−8.996513645720774300082880430682, −8.515938668734861799102336933105, −7.39981695496466471264079431099, −6.69244780946611109603682547232, −6.00565269769813463350540684999, −5.18833991148908023689652161330, −3.98041145897093721291724668118, −3.19527766147999966779548822191, −1.73524447107787891214306983391, −0.52587035887760909966628918424,
1.42838577536204313411444097810, 2.63664985203648139816597111801, 4.07592872819554934550152021802, 4.45197464767543214754913032764, 5.68474214831264826871796533582, 6.27410411663912612375961204904, 7.19415241444524821205014084566, 8.261425108835788191976865413006, 8.735155210672627422920302622228, 9.888138011829028749727984458184