L(s) = 1 | + (0.814 − 1.52i)3-s + (−2.08 − 2.08i)5-s − 1.14·7-s + (−1.67 − 2.48i)9-s + (−1.67 + 1.67i)11-s + (−0.146 − 0.146i)13-s + (−4.88 + 1.48i)15-s − 5.59i·17-s + (−1.48 + 1.48i)19-s + (−0.933 + 1.75i)21-s + 3.34i·23-s + 3.68i·25-s + (−5.16 + 0.533i)27-s + (3.51 − 3.51i)29-s − 5.83i·31-s + ⋯ |
L(s) = 1 | + (0.470 − 0.882i)3-s + (−0.931 − 0.931i)5-s − 0.433·7-s + (−0.558 − 0.829i)9-s + (−0.504 + 0.504i)11-s + (−0.0405 − 0.0405i)13-s + (−1.26 + 0.384i)15-s − 1.35i·17-s + (−0.341 + 0.341i)19-s + (−0.203 + 0.382i)21-s + 0.698i·23-s + 0.737i·25-s + (−0.994 + 0.102i)27-s + (0.652 − 0.652i)29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.252245 - 0.905397i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.252245 - 0.905397i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.814 + 1.52i)T \) |
good | 5 | \( 1 + (2.08 + 2.08i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + (1.67 - 1.67i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.146 + 0.146i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.59iT - 17T^{2} \) |
| 19 | \( 1 + (1.48 - 1.48i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.34iT - 23T^{2} \) |
| 29 | \( 1 + (-3.51 + 3.51i)T - 29iT^{2} \) |
| 31 | \( 1 + 5.83iT - 31T^{2} \) |
| 37 | \( 1 + (-4.83 + 4.83i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.610T + 41T^{2} \) |
| 43 | \( 1 + (-1.48 - 1.48i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 + (0.164 + 0.164i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.05 + 9.05i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.53 + 4.53i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.635 + 0.635i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.90iT - 71T^{2} \) |
| 73 | \( 1 + 7.07iT - 73T^{2} \) |
| 79 | \( 1 - 9.83iT - 79T^{2} \) |
| 83 | \( 1 + (-8.09 - 8.09i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.490T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−11.30649220125631692343772836345, −9.759478333270295814196608036105, −9.039197734371410348628101727690, −7.928917602065858895633503947760, −7.53863032695253235230963458187, −6.31401213712017284280230099877, −5.00219285093324934517258663671, −3.80261631814939999472667908199, −2.42684058434564036129609477098, −0.57960252942503700806024779729,
2.73759855601906288781256491800, 3.55802835446090136128909585342, 4.56533818726268321568573755416, 5.99229308267491378061193707492, 7.10649584301642006276172147168, 8.194167844906370733476762757282, 8.814025516642146743704812018158, 10.21340285262470598333645343492, 10.62219800298338189055517349116, 11.42104432045100858428432898245