| L(s) = 1 | + (−2.21 + 1.60i)2-s + (−2.44 − 7.54i)3-s + (−0.165 + 0.509i)4-s + (−12.0 − 8.73i)5-s + (17.5 + 12.7i)6-s + (0.949 − 2.92i)7-s + (−7.20 − 22.1i)8-s + (−29.0 + 21.0i)9-s + 40.5·10-s + 4.24·12-s + (−4.33 + 3.14i)13-s + (2.59 + 7.98i)14-s + (−36.3 + 112. i)15-s + (48.0 + 34.9i)16-s + (33.3 + 24.2i)17-s + (30.2 − 93.1i)18-s + ⋯ |
| L(s) = 1 | + (−0.781 + 0.567i)2-s + (−0.471 − 1.45i)3-s + (−0.0207 + 0.0637i)4-s + (−1.07 − 0.781i)5-s + (1.19 + 0.866i)6-s + (0.0512 − 0.157i)7-s + (−0.318 − 0.980i)8-s + (−1.07 + 0.780i)9-s + 1.28·10-s + 0.102·12-s + (−0.0924 + 0.0672i)13-s + (0.0495 + 0.152i)14-s + (−0.626 + 1.92i)15-s + (0.751 + 0.545i)16-s + (0.475 + 0.345i)17-s + (0.396 − 1.21i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0937835 + 0.117388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0937835 + 0.117388i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (2.21 - 1.60i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.44 + 7.54i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (12.0 + 8.73i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-0.949 + 2.92i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (4.33 - 3.14i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-33.3 - 24.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-43.2 - 133. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (7.72 - 23.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (25.4 - 18.5i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-4.06 + 12.5i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-80.6 - 248. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 57.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (106. + 327. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-277. + 201. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-27.3 + 84.0i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (597. + 434. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-167. - 121. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (312. - 961. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (1.04e3 - 760. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (357. + 259. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.08e3 - 791. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−12.86616635439403233722914635731, −12.31819793067250505686862148773, −11.64091412621198126555272000450, −9.919880657655484712997006859625, −8.351830902689068256864253355404, −7.932364025691760776064830730836, −7.07539549150320796984129851741, −5.82774946335877837845550251247, −3.88892692291587505519364114149, −1.20948828150126560026565782908,
0.12597261921877155317781230356, 2.97764375988463407796548786000, 4.38126842857172664856605856413, 5.62511967998897306588570240233, 7.44043964224867884320697950035, 8.833975963587008057750116806037, 9.760291948740097754811465328230, 10.60995290451484778428466732717, 11.30518671799113148713495114481, 11.92977024158741640413445202131
