L(s) = 1 | + (1.29 + 2.23i)2-s + (1.68 − 0.410i)3-s + (−2.34 + 4.05i)4-s + (0.5 − 0.866i)5-s + (3.09 + 3.23i)6-s + (2.13 + 3.70i)7-s − 6.95·8-s + (2.66 − 1.38i)9-s + 2.58·10-s + (−2.19 − 3.79i)11-s + (−2.27 + 7.79i)12-s + (0.5 − 0.866i)13-s + (−5.52 + 9.57i)14-s + (0.485 − 1.66i)15-s + (−4.30 − 7.44i)16-s + 0.619·17-s + ⋯ |
L(s) = 1 | + (0.914 + 1.58i)2-s + (0.971 − 0.237i)3-s + (−1.17 + 2.02i)4-s + (0.223 − 0.387i)5-s + (1.26 + 1.32i)6-s + (0.807 + 1.39i)7-s − 2.45·8-s + (0.887 − 0.461i)9-s + 0.817·10-s + (−0.660 − 1.14i)11-s + (−0.656 + 2.25i)12-s + (0.138 − 0.240i)13-s + (−1.47 + 2.55i)14-s + (0.125 − 0.429i)15-s + (−1.07 − 1.86i)16-s + 0.150·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39757 + 2.83102i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39757 + 2.83102i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.68 + 0.410i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1.29 - 2.23i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-2.13 - 3.70i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.19 + 3.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.619T + 17T^{2} \) |
| 19 | \( 1 + 2.61T + 19T^{2} \) |
| 23 | \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.47 + 6.02i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.44 + 5.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 + (-2.03 + 3.52i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.01 - 8.69i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.202 - 0.351i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + (-5.36 + 9.28i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.53 + 4.39i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.27 - 12.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6.57T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 + (7.27 + 12.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.52 - 2.63i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + (-4.81 - 8.34i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.31886572347399239218748894305, −9.570991272673117658968790406874, −8.745393359949861769701093240836, −8.082287332158941770898090707486, −7.74779374475475910112476311790, −6.19360202936910615211435500922, −5.70660877048268935330971913030, −4.73396413501297037763951827523, −3.56558073931073743370199406069, −2.33309427357688076210928844420,
1.53038100398638208290171981775, 2.41690708252113713300327897863, 3.61446248892015833869611747680, 4.39879083096349229176853171702, 5.04225234754303535022164173728, 6.83290285991297365402386803803, 7.79880810131217958377694623704, 8.987389546127545649230491402004, 10.07158092156799421311724320660, 10.48209180088947500895768467778