| L(s) = 1 | + (1.29 − 1.48i)2-s + (−0.250 − 1.84i)4-s + (3.26 − 1.06i)5-s + (−2.14 + 0.917i)7-s + (0.220 + 0.145i)8-s + (2.65 − 6.21i)10-s + (0.197 − 4.39i)11-s + (−2.83 + 0.127i)13-s + (−1.41 + 4.36i)14-s + (4.09 − 1.13i)16-s + (1.71 + 1.24i)17-s + (3.24 − 0.588i)19-s + (−2.78 − 5.77i)20-s + (−6.24 − 5.97i)22-s + (0.415 − 1.82i)23-s + ⋯ |
| L(s) = 1 | + (0.914 − 1.04i)2-s + (−0.125 − 0.924i)4-s + (1.46 − 0.474i)5-s + (−0.811 + 0.346i)7-s + (0.0778 + 0.0513i)8-s + (0.839 − 1.96i)10-s + (0.0594 − 1.32i)11-s + (−0.785 + 0.0352i)13-s + (−0.379 + 1.16i)14-s + (1.02 − 0.282i)16-s + (0.415 + 0.302i)17-s + (0.744 − 0.135i)19-s + (−0.621 − 1.29i)20-s + (−1.33 − 1.27i)22-s + (0.0867 − 0.379i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0889 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0889 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.91557 - 2.09436i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91557 - 2.09436i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (6.76 + 5.02i)T \) |
| good | 2 | \( 1 + (-1.29 + 1.48i)T + (-0.268 - 1.98i)T^{2} \) |
| 5 | \( 1 + (-3.26 + 1.06i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (2.14 - 0.917i)T + (4.83 - 5.05i)T^{2} \) |
| 11 | \( 1 + (-0.197 + 4.39i)T + (-10.9 - 0.986i)T^{2} \) |
| 13 | \( 1 + (2.83 - 0.127i)T + (12.9 - 1.16i)T^{2} \) |
| 17 | \( 1 + (-1.71 - 1.24i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-3.24 + 0.588i)T + (17.7 - 6.67i)T^{2} \) |
| 23 | \( 1 + (-0.415 + 1.82i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (6.35 - 6.07i)T + (1.30 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.326 + 1.18i)T + (-26.6 - 15.8i)T^{2} \) |
| 37 | \( 1 + (-0.00289 - 0.0126i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-2.50 - 3.14i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.686 - 0.410i)T + (20.3 - 37.8i)T^{2} \) |
| 47 | \( 1 + (-3.97 - 7.38i)T + (-25.8 + 39.2i)T^{2} \) |
| 53 | \( 1 + (1.13 - 8.40i)T + (-51.0 - 14.0i)T^{2} \) |
| 59 | \( 1 + (11.0 + 4.15i)T + (44.4 + 38.8i)T^{2} \) |
| 61 | \( 1 + (12.9 + 5.54i)T + (42.1 + 44.0i)T^{2} \) |
| 67 | \( 1 + (0.645 - 0.0874i)T + (64.5 - 17.8i)T^{2} \) |
| 73 | \( 1 + (-11.9 - 10.3i)T + (9.79 + 72.3i)T^{2} \) |
| 79 | \( 1 + (5.49 - 8.32i)T + (-31.0 - 72.6i)T^{2} \) |
| 83 | \( 1 + (-1.08 + 2.89i)T + (-62.5 - 54.6i)T^{2} \) |
| 89 | \( 1 + (-15.2 - 2.06i)T + (85.7 + 23.6i)T^{2} \) |
| 97 | \( 1 + (-1.66 - 1.32i)T + (21.5 + 94.5i)T^{2} \) |
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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
−10.51743274730747784868965245533, −9.547447222894012813186475742462, −9.116355217385061241153862039206, −7.73562741194520992083393040806, −6.21389937411380015726242239958, −5.64929754473593059083091709626, −4.81117584106377757385661333813, −3.36570637643166591016322181528, −2.64155678541548448210845736002, −1.36979760975976908463379044874,
1.97297667186214405396901798932, 3.33742894964826328814265459490, 4.63272615900497945585406518522, 5.53432073348663398620074967558, 6.21097686022380312922534251244, 7.14127697958454658464281797333, 7.51896380894207119749583958295, 9.383263787087961050672399513411, 9.847003402067345835713554500069, 10.45209134270764865227918881784
