L(s) = 1 | + (−1.39 + 0.258i)2-s + (1.86 − 0.719i)4-s + 0.732i·5-s + (−2.40 + 1.48i)8-s + (−0.189 − 1.01i)10-s − 2.03·11-s + 1.41·13-s + (2.96 − 2.68i)16-s − 4.19i·17-s + 5.56i·19-s + (0.526 + 1.36i)20-s + (2.83 − 0.526i)22-s − 2.03·23-s + 4.46·25-s + (−1.96 + 0.366i)26-s + ⋯ |
L(s) = 1 | + (−0.983 + 0.183i)2-s + (0.933 − 0.359i)4-s + 0.327i·5-s + (−0.851 + 0.524i)8-s + (−0.0599 − 0.321i)10-s − 0.613·11-s + 0.392·13-s + (0.741 − 0.671i)16-s − 1.01i·17-s + 1.27i·19-s + (0.117 + 0.305i)20-s + (0.603 − 0.112i)22-s − 0.424·23-s + 0.892·25-s + (−0.385 + 0.0717i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9095676180\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9095676180\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.258i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4.19iT - 17T^{2} \) |
| 19 | \( 1 - 5.56iT - 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 - 5.27iT - 29T^{2} \) |
| 31 | \( 1 + 9.63iT - 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 8.73iT - 41T^{2} \) |
| 43 | \( 1 - 7.86iT - 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.14iT - 53T^{2} \) |
| 59 | \( 1 + 4.98T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 - 2.87iT - 67T^{2} \) |
| 71 | \( 1 - 9.08T + 71T^{2} \) |
| 73 | \( 1 - 1.69T + 73T^{2} \) |
| 79 | \( 1 - 4.98iT - 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 8.73iT - 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.487694373972397164508117522514, −8.682246588070400662903998763393, −7.78943330135086035097096638995, −7.40668408015350157654921120868, −6.29566558464538474920452323613, −5.77046791442910205926396055848, −4.60744349551336833020271995318, −3.23418201477690477609403318519, −2.39550740066761767548613403791, −1.07481071144457209949236145061,
0.54860424030202857223233173013, 1.86556660497002907852933534240, 2.88363403081240800812658755149, 3.95346351916650061790230074990, 5.13566359291665462624461611345, 6.09727788581419588981480691799, 6.93320640760807165409080950199, 7.68439844868833018860868056194, 8.643016745904763379591575938949, 8.863158335702217050915072341276