| L(s) = 1 | + (0.142 − 0.989i)3-s + (−1.67 − 3.67i)5-s + (2.54 − 2.42i)7-s + (−0.959 − 0.281i)9-s + (5.23 + 0.499i)11-s + (−2.29 − 0.442i)13-s + (−3.87 + 1.13i)15-s + (−1.39 − 0.719i)17-s + (−0.391 − 0.373i)19-s + (−2.04 − 2.86i)21-s + (7.68 − 3.07i)23-s + (−7.41 + 8.55i)25-s + (−0.415 + 0.909i)27-s + (−0.551 + 0.955i)29-s + (−7.91 + 1.52i)31-s + ⋯ |
| L(s) = 1 | + (0.0821 − 0.571i)3-s + (−0.750 − 1.64i)5-s + (0.962 − 0.917i)7-s + (−0.319 − 0.0939i)9-s + (1.57 + 0.150i)11-s + (−0.636 − 0.122i)13-s + (−1.00 + 0.293i)15-s + (−0.338 − 0.174i)17-s + (−0.0897 − 0.0855i)19-s + (−0.445 − 0.625i)21-s + (1.60 − 0.641i)23-s + (−1.48 + 1.71i)25-s + (−0.0799 + 0.175i)27-s + (−0.102 + 0.177i)29-s + (−1.42 + 0.274i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.531472 - 1.38313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.531472 - 1.38313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-4.22 - 7.01i)T \) |
| good | 5 | \( 1 + (1.67 + 3.67i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 2.42i)T + (0.333 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-5.23 - 0.499i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (2.29 + 0.442i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (1.39 + 0.719i)T + (9.86 + 13.8i)T^{2} \) |
| 19 | \( 1 + (0.391 + 0.373i)T + (0.904 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-7.68 + 3.07i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (0.551 - 0.955i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.91 - 1.52i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-5.09 - 8.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.441 + 9.27i)T + (-40.8 + 3.89i)T^{2} \) |
| 43 | \( 1 + (7.98 - 5.12i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (3.79 + 2.98i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (4.24 + 2.72i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.79 - 3.22i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-4.29 + 0.410i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 6.74i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-6.99 + 0.667i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (2.12 + 6.13i)T + (-62.0 + 48.8i)T^{2} \) |
| 83 | \( 1 + (9.30 - 13.0i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-0.140 - 0.974i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.846 - 1.46i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.699479598240241102790977114948, −8.839980921237630826443132290545, −8.336141109797535568553882871135, −7.38663797045980850174330955552, −6.74452363661557548883865098041, −5.13145664454326002765209167693, −4.60058518935688367457276668281, −3.66878959093873526137590712069, −1.63914243175396313988492338619, −0.794103782976673237753681764524,
2.07670051203782222052563728467, 3.24058441095280527990144630940, 4.05643059742457512933120546380, 5.19016511075863170711186211370, 6.35063593756457492333269774969, 7.12355563479091662863545640674, 8.000820052605669245121728347559, 9.000752798599161144045869957627, 9.660065343518337510091586450964, 10.87177090623103433741688615052
