L(s) = 1 | + (0.988 + 0.149i)2-s + (0.520 − 0.354i)3-s + (0.955 + 0.294i)4-s + (−0.0807 − 1.07i)5-s + (0.567 − 0.273i)6-s + (−2.64 − 0.115i)7-s + (0.900 + 0.433i)8-s + (−0.951 + 2.42i)9-s + (0.0807 − 1.07i)10-s + (0.249 + 0.634i)11-s + (0.601 − 0.185i)12-s + (0.00686 − 0.00860i)13-s + (−2.59 − 0.507i)14-s + (−0.424 − 0.532i)15-s + (0.826 + 0.563i)16-s + (0.289 + 0.268i)17-s + ⋯ |
L(s) = 1 | + (0.699 + 0.105i)2-s + (0.300 − 0.204i)3-s + (0.477 + 0.147i)4-s + (−0.0361 − 0.482i)5-s + (0.231 − 0.111i)6-s + (−0.999 − 0.0435i)7-s + (0.318 + 0.153i)8-s + (−0.317 + 0.807i)9-s + (0.0255 − 0.340i)10-s + (0.0751 + 0.191i)11-s + (0.173 − 0.0535i)12-s + (0.00190 − 0.00238i)13-s + (−0.693 − 0.135i)14-s + (−0.109 − 0.137i)15-s + (0.206 + 0.140i)16-s + (0.0701 + 0.0650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42062 - 0.0467219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42062 - 0.0467219i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.64 + 0.115i)T \) |
good | 3 | \( 1 + (-0.520 + 0.354i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (0.0807 + 1.07i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.249 - 0.634i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.00686 + 0.00860i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.289 - 0.268i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.12 + 5.41i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.89 - 4.54i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.591 + 2.59i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.87 + 4.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.53 - 0.472i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-7.12 - 3.42i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.84 + 3.77i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (11.1 + 1.68i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-7.53 - 2.32i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.377 - 5.03i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 1.66i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (3.11 - 5.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.529 + 2.31i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.11 - 0.318i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (5.18 + 8.97i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.01 - 2.52i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (5.92 - 15.0i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 19.2T + 97T^{2} \) |
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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
−13.59956350854744481622306909054, −13.14686886248549498230813011633, −12.09365132724344609420964647352, −10.87357029861521985932253547499, −9.544502376676157143200625612121, −8.272206254575584040568553648709, −7.02094960583293323715219211454, −5.73162837018705308514023199133, −4.29946235734248983835968853248, −2.61114141782389390206647964347,
2.90182941408672796812239823819, 4.00174598202453897788633043892, 5.95628011681733755838344724050, 6.77875681010080725463191866791, 8.475564106707112246541376005081, 9.793315598140520143373849693028, 10.72867891348130689004770675710, 12.13644541396085454813501925919, 12.77038355622508684152012946662, 14.20385678826480242403968687695