| L(s) = 1 | + (−0.269 − 0.155i)2-s + (5.09 − 1.02i)3-s + (−3.95 − 6.84i)4-s + (14.5 + 3.89i)5-s + (−1.53 − 0.516i)6-s + (−12.0 + 3.24i)7-s + 4.94i·8-s + (24.9 − 10.4i)9-s + (−3.31 − 3.31i)10-s + (62.2 − 16.6i)11-s + (−27.1 − 30.8i)12-s + (−2.99 − 5.18i)13-s + (3.76 + 1.00i)14-s + (78.1 + 4.97i)15-s + (−30.8 + 53.4i)16-s + (−64.7 + 26.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.0952 − 0.0549i)2-s + (0.980 − 0.196i)3-s + (−0.493 − 0.855i)4-s + (1.30 + 0.348i)5-s + (−0.104 − 0.0351i)6-s + (−0.653 + 0.174i)7-s + 0.218i·8-s + (0.922 − 0.386i)9-s + (−0.104 − 0.104i)10-s + (1.70 − 0.457i)11-s + (−0.652 − 0.741i)12-s + (−0.0638 − 0.110i)13-s + (0.0718 + 0.0192i)14-s + (1.34 + 0.0855i)15-s + (−0.481 + 0.834i)16-s + (−0.923 + 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.21763 - 0.956876i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.21763 - 0.956876i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.09 + 1.02i)T \) |
| 17 | \( 1 + (64.7 - 26.9i)T \) |
| good | 2 | \( 1 + (0.269 + 0.155i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-14.5 - 3.89i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (12.0 - 3.24i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-62.2 + 16.6i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (2.99 + 5.18i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 95.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-26.4 + 98.8i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (-19.3 - 72.3i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-36.0 - 9.67i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-92.3 + 92.3i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (64.7 - 241. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-271. - 156. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (214. - 371. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 515. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (353. - 203. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (595. - 159. i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (-341. - 591. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (591. - 591. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (319. - 319. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (1.03e3 - 277. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (-324. - 187. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 797.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-267. - 996. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−12.86312540864018766726893963297, −11.12925656944403462652729407662, −9.977961446591904950274622726979, −9.256769890633141322870414985358, −8.766575096561317193845755667305, −6.66244692868400624337794469993, −6.19484513911619698222953466566, −4.40946523775555335385551200563, −2.70533658663349197146777389360, −1.32202839070519538333146074656,
1.81013607555525888774189405937, 3.43598110503930102730253566230, 4.51397517233209995515468566085, 6.33536203225369158398714475536, 7.46411516504092346347039529848, 8.921209019692790617986109876024, 9.309061084805585222882288656751, 10.06922092179872309328997876351, 11.94893199867503726710244561280, 12.90625465377734866040531316961
