| L(s) = 1 | + (−0.988 − 0.149i)2-s + (−0.934 + 0.637i)3-s + (0.955 + 0.294i)4-s + (0.0772 + 1.03i)5-s + (1.01 − 0.490i)6-s + (1.36 + 2.26i)7-s + (−0.900 − 0.433i)8-s + (−0.628 + 1.60i)9-s + (0.0772 − 1.03i)10-s + (1.68 + 4.29i)11-s + (−1.08 + 0.333i)12-s + (2.85 − 3.58i)13-s + (−1.01 − 2.44i)14-s + (−0.729 − 0.914i)15-s + (0.826 + 0.563i)16-s + (−4.77 − 4.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (−0.539 + 0.368i)3-s + (0.477 + 0.147i)4-s + (0.0345 + 0.460i)5-s + (0.416 − 0.200i)6-s + (0.516 + 0.856i)7-s + (−0.318 − 0.153i)8-s + (−0.209 + 0.533i)9-s + (0.0244 − 0.325i)10-s + (0.508 + 1.29i)11-s + (−0.312 + 0.0962i)12-s + (0.793 − 0.994i)13-s + (−0.271 − 0.653i)14-s + (−0.188 − 0.236i)15-s + (0.206 + 0.140i)16-s + (−1.15 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.559199 + 0.345827i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.559199 + 0.345827i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (-1.36 - 2.26i)T \) |
| good | 3 | \( 1 + (0.934 - 0.637i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.0772 - 1.03i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.68 - 4.29i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.85 + 3.58i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.77 + 4.43i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (0.366 + 0.634i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.65 - 1.53i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 5.71i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.69 - 2.93i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.26 + 1.00i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-6.33 - 3.05i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.51 + 1.69i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.70 - 0.407i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (4.46 + 1.37i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.433 + 5.78i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 0.421i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.84 + 8.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.41 - 14.9i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (6.08 - 0.917i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (7.53 + 13.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.09 + 3.88i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-6.46 + 16.4i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−14.31272456982249671416278206205, −12.82269714740584687335853683670, −11.58989547223138833610060371583, −10.97887896420268180771446665648, −9.874684696114288394577070061088, −8.743131598688546048203410012420, −7.50700989788672531157315920113, −6.11660284271417044826668884578, −4.72938580329621739058365693210, −2.42523381298414329842751538311,
1.17822009857824796620705529507, 4.03130717366155457878949442925, 5.99068838916977786405486186025, 6.83762440689102289035487537933, 8.418071475678360277637646184745, 9.089780699556716980789511045135, 10.88785987360265980526749572918, 11.24790964449003491776052358965, 12.55750879866980094054333839902, 13.75275564979479094550972012434
