| L(s) = 1 | + (−0.607 − 1.61i)2-s + (−0.742 + 0.648i)4-s + (−1.64 − 2.26i)5-s + (−1.56 + 1.03i)7-s + (−1.54 − 0.830i)8-s + (−2.66 + 4.03i)10-s + (−0.944 + 0.987i)11-s + (−1.36 + 1.30i)13-s + (2.61 + 1.90i)14-s + (−0.671 + 4.95i)16-s + (0.529 + 1.62i)17-s + (1.76 + 0.158i)19-s + (2.69 + 0.614i)20-s + (2.17 + 0.928i)22-s + (0.962 − 1.20i)23-s + ⋯ |
| L(s) = 1 | + (−0.429 − 1.14i)2-s + (−0.371 + 0.324i)4-s + (−0.735 − 1.01i)5-s + (−0.590 + 0.389i)7-s + (−0.545 − 0.293i)8-s + (−0.842 + 1.27i)10-s + (−0.284 + 0.297i)11-s + (−0.378 + 0.361i)13-s + (0.699 + 0.508i)14-s + (−0.167 + 1.23i)16-s + (0.128 + 0.394i)17-s + (0.404 + 0.0364i)19-s + (0.601 + 0.137i)20-s + (0.462 + 0.197i)22-s + (0.200 − 0.251i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0161487 + 0.0107595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0161487 + 0.0107595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (7.92 - 2.86i)T \) |
| good | 2 | \( 1 + (0.607 + 1.61i)T + (-1.50 + 1.31i)T^{2} \) |
| 5 | \( 1 + (1.64 + 2.26i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (1.56 - 1.03i)T + (2.75 - 6.43i)T^{2} \) |
| 11 | \( 1 + (0.944 - 0.987i)T + (-0.493 - 10.9i)T^{2} \) |
| 13 | \( 1 + (1.36 - 1.30i)T + (0.583 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.529 - 1.62i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 0.158i)T + (18.6 + 3.39i)T^{2} \) |
| 23 | \( 1 + (-0.962 + 1.20i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (3.27 - 1.39i)T + (20.0 - 20.9i)T^{2} \) |
| 31 | \( 1 + (3.17 - 0.430i)T + (29.8 - 8.24i)T^{2} \) |
| 37 | \( 1 + (-1.56 - 1.96i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (4.50 - 2.16i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (4.65 + 1.28i)T + (36.9 + 22.0i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 1.29i)T + (22.2 - 41.3i)T^{2} \) |
| 53 | \( 1 + (-4.70 - 4.10i)T + (7.11 + 52.5i)T^{2} \) |
| 59 | \( 1 + (8.54 - 1.55i)T + (55.2 - 20.7i)T^{2} \) |
| 61 | \( 1 + (-0.952 - 0.628i)T + (23.9 + 56.0i)T^{2} \) |
| 67 | \( 1 + (10.1 + 11.5i)T + (-8.99 + 66.3i)T^{2} \) |
| 73 | \( 1 + (-9.91 + 3.72i)T + (54.9 - 48.0i)T^{2} \) |
| 79 | \( 1 + (1.87 - 3.48i)T + (-43.5 - 65.9i)T^{2} \) |
| 83 | \( 1 + (0.853 + 4.70i)T + (-77.7 + 29.1i)T^{2} \) |
| 89 | \( 1 + (11.0 - 12.6i)T + (-11.9 - 88.1i)T^{2} \) |
| 97 | \( 1 + (-1.42 + 2.94i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.76253168191418577274959362057, −9.854988553022571676150496508759, −9.172045730829690589613268589027, −8.492621180359503473244342176035, −7.41818314825338669973566994098, −6.21432620854356642445113210309, −5.02392407719213177314052403596, −3.92933832415921475223900807371, −2.86783098073919200049990347656, −1.52333407902334339261773741233,
0.01194878100578040224495659552, 2.81313746340503311600516171914, 3.63583073390979207729600121455, 5.22812345500088798582113754259, 6.19860130457259368246254323695, 7.20127389574104344618871457344, 7.40863527039282438865230740215, 8.393362942461125816856791602683, 9.426474987691613979899274783724, 10.27849880952944553810735793091
