| L(s) = 1 | + (0.851 − 0.523i)2-s + (1.09 − 2.91i)3-s + (0.451 − 0.892i)4-s + (0.922 + 1.25i)5-s + (−0.596 − 3.05i)6-s + (−0.123 + 0.189i)7-s + (−0.0825 − 0.996i)8-s + (−5.03 − 4.38i)9-s + (1.44 + 0.585i)10-s + (2.00 + 0.687i)11-s + (−2.10 − 2.28i)12-s + (−2.24 − 2.72i)13-s + (−0.00622 + 0.225i)14-s + (4.66 − 1.31i)15-s + (−0.592 − 0.805i)16-s + (1.80 + 3.56i)17-s + ⋯ |
| L(s) = 1 | + (0.602 − 0.370i)2-s + (0.629 − 1.68i)3-s + (0.225 − 0.446i)4-s + (0.412 + 0.561i)5-s + (−0.243 − 1.24i)6-s + (−0.0467 + 0.0714i)7-s + (−0.0291 − 0.352i)8-s + (−1.67 − 1.46i)9-s + (0.456 + 0.185i)10-s + (0.603 + 0.207i)11-s + (−0.608 − 0.660i)12-s + (−0.623 − 0.756i)13-s + (−0.00166 + 0.0603i)14-s + (1.20 − 0.340i)15-s + (−0.148 − 0.201i)16-s + (0.438 + 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14744 - 2.38985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14744 - 2.38985i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.851 + 0.523i)T \) |
| 19 | \( 1 + (2.85 + 3.29i)T \) |
| good | 3 | \( 1 + (-1.09 + 2.91i)T + (-2.26 - 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.922 - 1.25i)T + (-1.49 + 4.77i)T^{2} \) |
| 7 | \( 1 + (0.123 - 0.189i)T + (-2.81 - 6.41i)T^{2} \) |
| 11 | \( 1 + (-2.00 - 0.687i)T + (8.68 + 6.75i)T^{2} \) |
| 13 | \( 1 + (2.24 + 2.72i)T + (-2.49 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-1.80 - 3.56i)T + (-10.0 + 13.6i)T^{2} \) |
| 23 | \( 1 + (1.90 + 5.09i)T + (-17.3 + 15.1i)T^{2} \) |
| 29 | \( 1 + (-8.28 - 0.457i)T + (28.8 + 3.19i)T^{2} \) |
| 31 | \( 1 + (7.60 - 4.11i)T + (16.9 - 25.9i)T^{2} \) |
| 37 | \( 1 + (-9.38 - 3.22i)T + (29.1 + 22.7i)T^{2} \) |
| 41 | \( 1 + (-6.44 - 6.27i)T + (1.12 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.435 + 0.176i)T + (30.8 - 29.9i)T^{2} \) |
| 47 | \( 1 + (-2.27 - 11.6i)T + (-43.5 + 17.6i)T^{2} \) |
| 53 | \( 1 + (0.213 + 1.09i)T + (-49.1 + 19.9i)T^{2} \) |
| 59 | \( 1 + (0.621 + 0.604i)T + (1.62 + 58.9i)T^{2} \) |
| 61 | \( 1 + (1.30 - 0.615i)T + (38.7 - 47.0i)T^{2} \) |
| 67 | \( 1 + (9.25 - 6.41i)T + (23.4 - 62.7i)T^{2} \) |
| 71 | \( 1 + (-2.74 - 1.29i)T + (45.1 + 54.8i)T^{2} \) |
| 73 | \( 1 + (-0.105 - 0.208i)T + (-43.2 + 58.8i)T^{2} \) |
| 79 | \( 1 + (-12.9 + 5.27i)T + (56.6 - 55.0i)T^{2} \) |
| 83 | \( 1 + (5.41 + 12.3i)T + (-56.2 + 61.0i)T^{2} \) |
| 89 | \( 1 + (2.62 - 5.19i)T + (-52.7 - 71.7i)T^{2} \) |
| 97 | \( 1 + (0.937 + 0.650i)T + (34.0 + 90.8i)T^{2} \) |
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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
−10.25559125721789188605734909327, −9.147041151059084396081093875400, −8.195525768238890996766311358997, −7.37887155303229570098156679388, −6.38877474429337934483762445710, −6.10538781450062463780523651826, −4.47598626810292797984329611853, −2.93838041658367230667169698623, −2.43120059328291134076113074328, −1.13808383572557431749186613013,
2.29751878721659511600591385865, 3.61111670857762713708405207749, 4.24411808224386094898470040079, 5.15523110979255622830696914871, 5.87085188977729136695555710238, 7.27879046306366614616724906332, 8.322616186694953538689293685127, 9.295714965489445759492209140210, 9.523748820109907774022877153982, 10.57635814349724491839530842197
