L(s) = 1 | + (0.328 − 3.98i)2-s + (−15.7 − 2.61i)4-s − 5.66·5-s − 52.1i·7-s + (−15.6 + 62.0i)8-s + (−1.85 + 22.5i)10-s + 106. i·11-s − 122.·13-s + (−207. − 17.1i)14-s + (242. + 82.6i)16-s + 122.·17-s + 593. i·19-s + (89.3 + 14.8i)20-s + (425. + 35.0i)22-s − 546. i·23-s + ⋯ |
L(s) = 1 | + (0.0820 − 0.996i)2-s + (−0.986 − 0.163i)4-s − 0.226·5-s − 1.06i·7-s + (−0.244 + 0.969i)8-s + (−0.0185 + 0.225i)10-s + 0.881i·11-s − 0.722·13-s + (−1.06 − 0.0873i)14-s + (0.946 + 0.322i)16-s + 0.424·17-s + 1.64i·19-s + (0.223 + 0.0370i)20-s + (0.878 + 0.0723i)22-s − 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.216625967\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.216625967\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.328 + 3.98i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 5.66T + 625T^{2} \) |
| 7 | \( 1 + 52.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 106. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 122.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 122.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 593. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 546. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 735.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 585. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.28e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.86e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.24e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.75e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.15e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 66.3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.10e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 5.03e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.70e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.38e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.00e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.18e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 4.81e3T + 8.85e7T^{2} \) |
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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
−10.86877341263773096506908193514, −10.09324664514895675798271073564, −9.526007236270379521308834800867, −8.079277245458877420066723085593, −7.37517352915212696385288518911, −5.82215760527688793048051846209, −4.47725343283274126446831039188, −3.82885337012658591817260011196, −2.34138684902191240460494630201, −1.00246280131063610279797108478,
0.44556546546620039477339252629, 2.65306890790590029806897270373, 4.03955558164876065744437145598, 5.35418175924442713428761314066, 5.93848291301766152401258815705, 7.23485547637445085067073116002, 8.019896194454010819744103667958, 9.099408236082279131903611741954, 9.578664926974631035262322781286, 11.15887575093082156720965824001