L(s) = 1 | + (−1 − 1.73i)5-s + (0.5 + 2.59i)7-s − 2·13-s + (3 − 5.19i)17-s + (0.5 + 0.866i)19-s + (−1 − 1.73i)23-s + (0.500 − 0.866i)25-s + 6·29-s + (−0.5 + 0.866i)31-s + (4 − 3.46i)35-s + (−1 − 1.73i)37-s + 2·41-s + 9·43-s + (1 + 1.73i)47-s + (−6.5 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.188 + 0.981i)7-s − 0.554·13-s + (0.727 − 1.26i)17-s + (0.114 + 0.198i)19-s + (−0.208 − 0.361i)23-s + (0.100 − 0.173i)25-s + 1.11·29-s + (−0.0898 + 0.155i)31-s + (0.676 − 0.585i)35-s + (−0.164 − 0.284i)37-s + 0.312·41-s + 1.37·43-s + (0.145 + 0.252i)47-s + (−0.928 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.446534722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.446534722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 9T + 43T^{2} \) |
| 47 | \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−9.328452850962182473162756689369, −8.516394905834565590537238602637, −7.930409222946531544087263450389, −7.02293442072793526739936197948, −5.94503026095327874346981430824, −5.09134814278855467995815352418, −4.53062754939942534503311676295, −3.20222581647745350566525667192, −2.21503541101929305742060862499, −0.67869172730049211863501936009,
1.14381699843739766241159282098, 2.64211136845691535474448416617, 3.66675731695643390722182114447, 4.34679744044240182802171336437, 5.50816857353372588428262268864, 6.48800632249584503681377331316, 7.31684744172923089916144354756, 7.76348952180550052799532236895, 8.694665748065900063379766906093, 9.798295131946900679255190336462