L(s) = 1 | + (0.171 − 0.252i)2-s + (4.85 + 1.85i)3-s + (2.92 + 7.34i)4-s + (−2.63 + 5.69i)5-s + (1.29 − 0.908i)6-s + (4.19 + 15.0i)7-s + (4.73 + 1.04i)8-s + (20.1 + 17.9i)9-s + (0.985 + 1.63i)10-s + (−43.2 − 40.9i)11-s + (0.611 + 41.0i)12-s + (−1.84 + 16.9i)13-s + (4.52 + 1.52i)14-s + (−23.3 + 22.7i)15-s + (−44.8 + 42.4i)16-s + (65.8 + 18.2i)17-s + ⋯ |
L(s) = 1 | + (0.0605 − 0.0892i)2-s + (0.934 + 0.356i)3-s + (0.365 + 0.918i)4-s + (−0.235 + 0.508i)5-s + (0.0883 − 0.0618i)6-s + (0.226 + 0.815i)7-s + (0.209 + 0.0460i)8-s + (0.746 + 0.665i)9-s + (0.0311 + 0.0518i)10-s + (−1.18 − 1.12i)11-s + (0.0147 + 0.988i)12-s + (−0.0394 + 0.362i)13-s + (0.0864 + 0.0291i)14-s + (−0.401 + 0.391i)15-s + (−0.700 + 0.663i)16-s + (0.939 + 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.56341 + 1.79772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56341 + 1.79772i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.85 - 1.85i)T \) |
| 59 | \( 1 + (-152. + 426. i)T \) |
good | 2 | \( 1 + (-0.171 + 0.252i)T + (-2.96 - 7.43i)T^{2} \) |
| 5 | \( 1 + (2.63 - 5.69i)T + (-80.9 - 95.2i)T^{2} \) |
| 7 | \( 1 + (-4.19 - 15.0i)T + (-293. + 176. i)T^{2} \) |
| 11 | \( 1 + (43.2 + 40.9i)T + (72.0 + 1.32e3i)T^{2} \) |
| 13 | \( 1 + (1.84 - 16.9i)T + (-2.14e3 - 472. i)T^{2} \) |
| 17 | \( 1 + (-65.8 - 18.2i)T + (4.20e3 + 2.53e3i)T^{2} \) |
| 19 | \( 1 + (80.5 + 61.2i)T + (1.83e3 + 6.60e3i)T^{2} \) |
| 23 | \( 1 + (9.89 + 18.6i)T + (-6.82e3 + 1.00e4i)T^{2} \) |
| 29 | \( 1 + (-234. + 158. i)T + (9.02e3 - 2.26e4i)T^{2} \) |
| 31 | \( 1 + (135. + 178. i)T + (-7.96e3 + 2.87e4i)T^{2} \) |
| 37 | \( 1 + (-62.9 - 286. i)T + (-4.59e4 + 2.12e4i)T^{2} \) |
| 41 | \( 1 + (-383. - 203. i)T + (3.86e4 + 5.70e4i)T^{2} \) |
| 43 | \( 1 + (-377. - 398. i)T + (-4.30e3 + 7.93e4i)T^{2} \) |
| 47 | \( 1 + (290. - 134. i)T + (6.72e4 - 7.91e4i)T^{2} \) |
| 53 | \( 1 + (77.6 - 129. i)T + (-6.97e4 - 1.31e5i)T^{2} \) |
| 61 | \( 1 + (59.7 + 40.5i)T + (8.40e4 + 2.10e5i)T^{2} \) |
| 67 | \( 1 + (-204. + 927. i)T + (-2.72e5 - 1.26e5i)T^{2} \) |
| 71 | \( 1 + (204. + 442. i)T + (-2.31e5 + 2.72e5i)T^{2} \) |
| 73 | \( 1 + (-132. + 394. i)T + (-3.09e5 - 2.35e5i)T^{2} \) |
| 79 | \( 1 + (7.71 - 142. i)T + (-4.90e5 - 5.33e4i)T^{2} \) |
| 83 | \( 1 + (-19.6 + 119. i)T + (-5.41e5 - 1.82e5i)T^{2} \) |
| 89 | \( 1 + (417. + 615. i)T + (-2.60e5 + 6.54e5i)T^{2} \) |
| 97 | \( 1 + (-486. - 1.44e3i)T + (-7.26e5 + 5.52e5i)T^{2} \) |
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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
−12.65531802613838473595197645307, −11.38674384316828216919884389986, −10.69958427078602003274609254837, −9.305276655030502911742716652922, −8.177057271950045251083222229807, −7.82045420457228662911751879746, −6.26509974313497936182893230330, −4.56970027271400843275988788269, −3.13834106635304570398808524469, −2.48899187785744037364216350145,
0.988632629997626934689409655173, 2.39655952685554569047111424131, 4.20680035163664071094438482873, 5.39675477099421848429402503593, 7.02827834370866012275897978290, 7.67607656515864840554945245722, 8.840783524319402677653563769825, 10.20749764011709744148548093572, 10.51314633638196008622278992992, 12.35101039250260144640574768849