| L(s) = 1 | + (1.26 + 0.626i)2-s + (2.47 − 1.24i)3-s + (1.21 + 1.58i)4-s + (−1.21 − 2.73i)5-s + (3.91 − 0.0294i)6-s + (0.284 + 0.170i)7-s + (0.546 + 2.77i)8-s + (2.78 − 3.75i)9-s + (0.176 − 4.23i)10-s + (1.52 + 1.19i)11-s + (4.98 + 2.41i)12-s + (−4.75 + 4.52i)13-s + (0.253 + 0.394i)14-s + (−6.41 − 5.26i)15-s + (−1.04 + 3.86i)16-s + (1.34 + 1.63i)17-s + ⋯ |
| L(s) = 1 | + (0.896 + 0.442i)2-s + (1.42 − 0.718i)3-s + (0.607 + 0.794i)4-s + (−0.542 − 1.22i)5-s + (1.59 − 0.0120i)6-s + (0.107 + 0.0644i)7-s + (0.193 + 0.981i)8-s + (0.927 − 1.25i)9-s + (0.0556 − 1.33i)10-s + (0.460 + 0.359i)11-s + (1.43 + 0.697i)12-s + (−1.31 + 1.25i)13-s + (0.0678 + 0.105i)14-s + (−1.65 − 1.35i)15-s + (−0.261 + 0.965i)16-s + (0.325 + 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.34334 - 0.360239i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.34334 - 0.360239i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.26 - 0.626i)T \) |
| good | 3 | \( 1 + (-2.47 + 1.24i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.21 + 2.73i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-0.284 - 0.170i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-1.52 - 1.19i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (4.75 - 4.52i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.34 - 1.63i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (1.37 + 7.93i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (1.57 + 0.742i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 1.14i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-1.81 + 0.360i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (2.43 + 1.54i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 1.62i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.11 - 9.43i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (0.771 + 0.412i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (10.0 - 5.67i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (8.88 - 9.32i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.0915 - 1.24i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (7.61 + 6.56i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-5.21 - 0.772i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-4.84 + 2.90i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-14.9 - 4.53i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (9.17 - 5.81i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-13.8 + 6.54i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (4.60 + 3.07i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.31313866497609713389904911424, −9.433957962260844282681689922447, −8.874703698456851029930080337291, −8.039973854154575881713332822518, −7.31643453662705959343964189879, −6.52514905658279817202192402113, −4.76913081799627948071827155227, −4.33511028001381144435385868637, −2.90960643740381484446507872567, −1.80397032135805345553090578724,
2.27716566782555085931520986719, 3.26326797627914270160961740703, 3.67561339968673550898340998753, 4.91498318701861389264724226741, 6.23298906158798693331300029962, 7.50580077332880794284232209602, 8.019530735925343796427740558355, 9.492392891324174080574888020244, 10.22435061757152534764121628013, 10.68043654013320821602261078664
