L(s) = 1 | + (−0.0218 − 1.41i)2-s + (−1.99 + 0.0617i)4-s + (−0.317 + 0.550i)5-s + (2.11 + 1.58i)7-s + (0.130 + 2.82i)8-s + (0.785 + 0.437i)10-s + (−3.16 + 1.82i)11-s + 4.15i·13-s + (2.19 − 3.03i)14-s + (3.99 − 0.246i)16-s + (−3.01 + 1.74i)17-s + (−1.99 + 3.45i)19-s + (0.601 − 1.11i)20-s + (2.65 + 4.43i)22-s + (1.47 − 2.54i)23-s + ⋯ |
L(s) = 1 | + (−0.0154 − 0.999i)2-s + (−0.999 + 0.0308i)4-s + (−0.142 + 0.246i)5-s + (0.800 + 0.598i)7-s + (0.0462 + 0.998i)8-s + (0.248 + 0.138i)10-s + (−0.954 + 0.550i)11-s + 1.15i·13-s + (0.586 − 0.810i)14-s + (0.998 − 0.0616i)16-s + (−0.732 + 0.422i)17-s + (−0.457 + 0.791i)19-s + (0.134 − 0.250i)20-s + (0.565 + 0.945i)22-s + (0.306 − 0.530i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05154 + 0.225775i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05154 + 0.225775i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0218 + 1.41i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.11 - 1.58i)T \) |
good | 5 | \( 1 + (0.317 - 0.550i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.16 - 1.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.15iT - 13T^{2} \) |
| 17 | \( 1 + (3.01 - 1.74i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.99 - 3.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.47 + 2.54i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + (-5.20 + 3.00i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.59 - 0.923i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 + (4.61 - 7.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.99 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (9.10 - 5.25i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.72 - 0.995i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.01 + 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.737T + 71T^{2} \) |
| 73 | \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.74 + 4.47i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.67iT - 83T^{2} \) |
| 89 | \( 1 + (-4.63 - 2.67i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−10.91381550456392261757185133123, −10.38544755770342203770363933577, −9.232936647288629898712472431099, −8.514639280660464882702602766692, −7.63216549046592216443782604947, −6.23774829367822249004841241298, −4.93353499943215995851389504771, −4.27632835837269149517299254577, −2.73143633472960345625005024094, −1.77487966770591596682664002828,
0.67296185522815811146779928128, 3.02579065916977166668750649832, 4.61019849178110081363889112634, 5.06387381301588685054106754152, 6.31783503171305201254373796486, 7.29508427411346804449959299279, 8.253949783892368078863408727745, 8.557744288978557597961159057280, 9.999729120279803340960933332938, 10.66005250381718153858360587045