L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 0.999i·6-s + (−1.41 − 2.45i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s − 0.999·10-s + 1.83·11-s + (0.499 − 0.866i)12-s + (0.486 − 0.281i)13-s − 2.83i·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s + (0.254 + 0.146i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s − 0.408i·6-s + (−0.535 − 0.927i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s − 0.316·10-s + 0.552·11-s + (0.144 − 0.249i)12-s + (0.135 − 0.0779i)13-s − 0.757i·14-s + (0.223 + 0.129i)15-s + (−0.125 + 0.216i)16-s + (0.0617 + 0.0356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.624316000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.624316000\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (4.07 + 4.51i)T \) |
good | 7 | \( 1 + (1.41 + 2.45i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + (-0.486 + 0.281i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.254 - 0.146i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.71 + 1.56i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.04iT - 23T^{2} \) |
| 29 | \( 1 + 0.994iT - 29T^{2} \) |
| 31 | \( 1 + 10.1iT - 31T^{2} \) |
| 41 | \( 1 + (-1.81 - 3.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 4.77iT - 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 + (-2.65 + 4.60i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.33 - 2.50i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.14 + 1.81i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.43 - 5.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.91 - 8.51i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.07T + 73T^{2} \) |
| 79 | \( 1 + (10.8 - 6.28i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.84 + 8.38i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.70 - 3.87i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−9.802534621962044251144277897495, −8.654335858524481118224883588197, −7.76676226207890318699309004691, −6.99092882115218169810716266789, −6.50931912033933492211903106486, −5.53588175816730852405594636684, −4.35066520076095205583841464306, −3.66591944773408579770351561426, −2.43870745391951365043491337073, −0.63820763070123464610956884943,
1.48073790798307185108225835832, 3.08447970324205513699569391331, 3.68075532214318349879693819972, 4.87320765635407749789723136648, 5.54947708740298245186315353978, 6.38192792030886537194155419442, 7.35917505320463587652785070569, 8.599916698747562196904908114396, 9.307170196613925208249178493923, 9.980458307236010831187790465930