L(s) = 1 | + (0.893 − 1.54i)2-s + (−0.5 + 0.866i)3-s + (−0.596 − 1.03i)4-s − 0.535·5-s + (0.893 + 1.54i)6-s + (−2.30 − 4.00i)7-s + 1.44·8-s + (−0.499 − 0.866i)9-s + (−0.478 + 0.828i)10-s + (0.5 − 0.866i)11-s + 1.19·12-s + (1.10 − 3.43i)13-s − 8.25·14-s + (0.267 − 0.463i)15-s + (2.48 − 4.29i)16-s + (−2.01 − 3.48i)17-s + ⋯ |
L(s) = 1 | + (0.631 − 1.09i)2-s + (−0.288 + 0.499i)3-s + (−0.298 − 0.516i)4-s − 0.239·5-s + (0.364 + 0.631i)6-s + (−0.873 − 1.51i)7-s + 0.510·8-s + (−0.166 − 0.288i)9-s + (−0.151 + 0.261i)10-s + (0.150 − 0.261i)11-s + 0.344·12-s + (0.306 − 0.951i)13-s − 2.20·14-s + (0.0691 − 0.119i)15-s + (0.620 − 1.07i)16-s + (−0.487 − 0.844i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.732960 - 1.35584i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.732960 - 1.35584i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-1.10 + 3.43i)T \) |
good | 2 | \( 1 + (-0.893 + 1.54i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.535T + 5T^{2} \) |
| 7 | \( 1 + (2.30 + 4.00i)T + (-3.5 + 6.06i)T^{2} \) |
| 17 | \( 1 + (2.01 + 3.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 6.50i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.59 + 4.49i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 4.17i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + (1.99 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.70 - 8.15i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.90 - 6.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.31T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + (2.07 + 3.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.87 - 8.45i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.91 - 6.77i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.18 + 8.98i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.96T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.95 + 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.84153052521485128308859830182, −10.24061106879311022428638020516, −9.639368800653042881821067443321, −8.029779708074334360761521035798, −7.11888564277233345256275437494, −5.86429573105328217546581628879, −4.54664674471321447733554884090, −3.75217313228270146566772201052, −3.00815892927743576940307678187, −0.870127760368795733359959870770,
2.09352299535168126887637281681, 3.75247959900953949793907612644, 5.18263584690916710978702469556, 5.82164400169444378165228400232, 6.76542887564344017107013601900, 7.29803117578175410413597845955, 8.717860452729443731183991775848, 9.267135539858140620471250767003, 10.75543897736799951446535046877, 11.75606603129439762074779438814