| 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
−13.75275564979479094550972012434, −12.55750879866980094054333839902, −11.24790964449003491776052358965, −10.88785987360265980526749572918, −9.089780699556716980789511045135, −8.418071475678360277637646184745, −6.83762440689102289035487537933, −5.99068838916977786405486186025, −4.03130717366155457878949442925, −1.17822009857824796620705529507,
2.42523381298414329842751538311, 4.72938580329621739058365693210, 6.11660284271417044826668884578, 7.50700989788672531157315920113, 8.743131598688546048203410012420, 9.874684696114288394577070061088, 10.97887896420268180771446665648, 11.58989547223138833610060371583, 12.82269714740584687335853683670, 14.31272456982249671416278206205
