| L(s) = 1 | + (−0.844 + 2.59i)2-s + (6.41 − 4.66i)3-s + (0.433 + 0.314i)4-s + (4.59 + 14.1i)5-s + (6.69 + 20.6i)6-s + (2.48 + 1.80i)7-s + (−18.8 + 13.7i)8-s + (11.0 − 34.1i)9-s − 40.5·10-s + 4.24·12-s + (−1.65 + 5.09i)13-s + (−6.78 + 4.93i)14-s + (95.2 + 69.2i)15-s + (−18.3 − 56.5i)16-s + (12.7 + 39.1i)17-s + (79.2 + 57.5i)18-s + ⋯ |
| L(s) = 1 | + (−0.298 + 0.918i)2-s + (1.23 − 0.896i)3-s + (0.0541 + 0.0393i)4-s + (0.410 + 1.26i)5-s + (0.455 + 1.40i)6-s + (0.134 + 0.0974i)7-s + (−0.833 + 0.605i)8-s + (0.410 − 1.26i)9-s − 1.28·10-s + 0.102·12-s + (−0.0353 + 0.108i)13-s + (−0.129 + 0.0941i)14-s + (1.64 + 1.19i)15-s + (−0.286 − 0.883i)16-s + (0.181 + 0.559i)17-s + (1.03 + 0.753i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.75360 + 1.47061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75360 + 1.47061i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.844 - 2.59i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (-6.41 + 4.66i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-4.59 - 14.1i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-2.48 - 1.80i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (1.65 - 5.09i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-12.7 - 39.1i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-113. + 82.2i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (20.2 + 14.6i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-9.73 + 29.9i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (10.6 + 7.72i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-211. + 153. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 57.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-278. + 202. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (105. - 326. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (71.4 + 51.9i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (228. + 702. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (64.0 + 197. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (817. + 594. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (399. - 1.23e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (136. + 420. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-416. + 1.28e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−13.73326819487245977854561853239, −12.35480825400315733407369752078, −11.16443251721922521446356737824, −9.677356714303755745258212665914, −8.574235678725818898104992749951, −7.55972422100474842526870045013, −6.99963929244320940939876340446, −5.91552609649414497742919068880, −3.20762335356387909172012697322, −2.21761131438765140415189701665,
1.35269043577622724563219000577, 2.87232626338767895051587389794, 4.18016193629570940945569556239, 5.62775085594744691913975765990, 7.82425681703682971888445358938, 8.961973570906199714931218829198, 9.598133045269660617261370650272, 10.27068042607962909561975507314, 11.69646976179848383248269099222, 12.64631143432763894300286981305
