| L(s) = 1 | + (1.26 − 0.624i)2-s + (0.743 − 1.56i)3-s + (1.21 − 1.58i)4-s + (−2.14 − 2.44i)5-s + (−0.0350 − 2.44i)6-s + (3.97 + 0.523i)7-s + (0.555 − 2.77i)8-s + (−1.89 − 2.32i)9-s + (−4.24 − 1.75i)10-s + (2.21 + 4.49i)11-s + (−1.57 − 3.08i)12-s + (1.12 + 3.30i)13-s + (5.36 − 1.82i)14-s + (−5.41 + 1.53i)15-s + (−1.02 − 3.86i)16-s + (−3.35 − 3.35i)17-s + ⋯ |
| L(s) = 1 | + (0.897 − 0.441i)2-s + (0.429 − 0.903i)3-s + (0.609 − 0.792i)4-s + (−0.957 − 1.09i)5-s + (−0.0143 − 0.999i)6-s + (1.50 + 0.197i)7-s + (0.196 − 0.980i)8-s + (−0.631 − 0.775i)9-s + (−1.34 − 0.556i)10-s + (0.668 + 1.35i)11-s + (−0.454 − 0.890i)12-s + (0.310 + 0.916i)13-s + (1.43 − 0.486i)14-s + (−1.39 + 0.396i)15-s + (−0.257 − 0.966i)16-s + (−0.812 − 0.812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39776 - 2.37648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39776 - 2.37648i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 + 0.624i)T \) |
| 3 | \( 1 + (-0.743 + 1.56i)T \) |
| good | 5 | \( 1 + (2.14 + 2.44i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-3.97 - 0.523i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 4.49i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.12 - 3.30i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (3.35 + 3.35i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.689 - 3.46i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.198 - 1.51i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (1.86 - 0.122i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (5.71 - 3.30i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.865 - 4.34i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.763 + 5.80i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-8.98 + 4.42i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-10.5 + 2.83i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (5.62 - 8.42i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-2.57 - 2.94i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.320 - 4.88i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-0.408 - 0.201i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-4.11 - 9.94i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-4.00 + 9.66i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (1.60 + 5.99i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (7.06 + 6.19i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-2.92 + 1.20i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (2.93 + 1.69i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.92663754784842171353113357757, −9.276153391824974578990926430825, −8.719011149438943145173522353128, −7.54518474716882509725273081976, −7.03287437561991340354421165476, −5.57470575577610650503753852018, −4.52015308752986854396071751855, −4.00230415144114996254080519528, −2.10230400061723201834648655751, −1.35087989511617367834432721007,
2.54504095482953437028000014949, 3.66084974972867732299757274007, 4.17715763257643483827678074582, 5.33342780191409584308863282586, 6.36652630388461259952911252971, 7.63137806458289311433063830159, 8.132476754673593280299270052274, 8.909543971819881526204399866346, 10.78578268837627134031479646596, 11.16372066882453241401713501408
