L(s) = 1 | + (−3.20 + 5.54i)3-s + (−9.34 + 16.1i)5-s − 16.3·7-s + (−7.01 − 12.1i)9-s + 53.5·11-s + (−42.4 − 73.5i)13-s + (−59.8 − 103. i)15-s + (−24.0 + 41.5i)17-s + (80.5 + 19.1i)19-s + (52.2 − 90.5i)21-s + (55.9 + 96.8i)23-s + (−111. − 193. i)25-s − 83.0·27-s + (−131. − 227. i)29-s − 122.·31-s + ⋯ |
L(s) = 1 | + (−0.616 + 1.06i)3-s + (−0.835 + 1.44i)5-s − 0.880·7-s + (−0.259 − 0.450i)9-s + 1.46·11-s + (−0.906 − 1.56i)13-s + (−1.02 − 1.78i)15-s + (−0.342 + 0.593i)17-s + (0.972 + 0.231i)19-s + (0.543 − 0.940i)21-s + (0.507 + 0.878i)23-s + (−0.895 − 1.55i)25-s − 0.592·27-s + (−0.840 − 1.45i)29-s − 0.708·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.156566 - 0.277661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156566 - 0.277661i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-80.5 - 19.1i)T \) |
good | 3 | \( 1 + (3.20 - 5.54i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (9.34 - 16.1i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 11 | \( 1 - 53.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + (42.4 + 73.5i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (24.0 - 41.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 23 | \( 1 + (-55.9 - 96.8i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (131. + 227. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + (207. - 359. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (29.2 - 50.6i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (85.2 + 147. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-165. - 287. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-354. + 613. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-8.95 - 15.5i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-164. - 285. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (31.4 - 54.4i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (281. - 487. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (154. - 266. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 524.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (298. + 516. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-275. + 476. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.09815901937935005275268198774, −11.75408858585301542791026341997, −11.19949511392723976146561322374, −10.11324950667356859672770026863, −9.622434762248073582922705288065, −7.76879661407297267042342294634, −6.75100040057492582350802773521, −5.59292085709248664915037726079, −3.94582057062797977588807342984, −3.18344663352096070934216950970,
0.17378942818915227529490669437, 1.46476850908603937665224022157, 3.92269903628279132438640292823, 5.16442679385987294131804086684, 6.75903162616693257444396516950, 7.20391748210400753446565413352, 8.946235227844610732055059506529, 9.349187577792674429871427615361, 11.46218467394019592195789922122, 12.04036003686982510776238652752