L(s) = 1 | + (−0.984 + 0.173i)2-s + (1.67 + 0.441i)3-s + (0.939 − 0.342i)4-s + (−1.90 − 1.16i)5-s + (−1.72 − 0.144i)6-s + (−2.42 − 1.40i)7-s + (−0.866 + 0.5i)8-s + (2.60 + 1.48i)9-s + (2.08 + 0.819i)10-s + (−5.08 + 2.93i)11-s + (1.72 − 0.157i)12-s + (−2.44 − 2.05i)13-s + (2.63 + 0.959i)14-s + (−2.67 − 2.79i)15-s + (0.766 − 0.642i)16-s + (0.740 + 4.20i)17-s + ⋯ |
L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.966 + 0.255i)3-s + (0.469 − 0.171i)4-s + (−0.852 − 0.522i)5-s + (−0.704 − 0.0589i)6-s + (−0.918 − 0.530i)7-s + (−0.306 + 0.176i)8-s + (0.869 + 0.493i)9-s + (0.657 + 0.259i)10-s + (−1.53 + 0.884i)11-s + (0.497 − 0.0454i)12-s + (−0.678 − 0.569i)13-s + (0.704 + 0.256i)14-s + (−0.691 − 0.722i)15-s + (0.191 − 0.160i)16-s + (0.179 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0265369 + 0.206206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0265369 + 0.206206i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-1.67 - 0.441i)T \) |
| 5 | \( 1 + (1.90 + 1.16i)T \) |
| 19 | \( 1 + (1.47 - 4.10i)T \) |
good | 7 | \( 1 + (2.42 + 1.40i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.08 - 2.93i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 2.05i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.740 - 4.20i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (7.84 - 2.85i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.140 + 0.794i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.327 + 0.189i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.577T + 37T^{2} \) |
| 41 | \( 1 + (-6.13 + 5.14i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.0911 + 0.250i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.53 + 8.68i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.39 - 3.84i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.97 + 11.2i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (5.56 - 2.02i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.0547 + 0.310i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (11.7 + 4.29i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.930 + 1.10i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.149 - 0.178i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.81 - 3.15i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.3 + 9.50i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.19 - 12.4i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.55846167471776920697084329575, −10.19069239932313842971889888049, −9.476821306886798642255769534357, −8.250521820772733231284762624744, −7.83381007577314726686072768934, −7.18723047314755178737595399847, −5.65676382408642375343931595403, −4.32475856416333309974597579558, −3.39509086626318826877439531073, −2.06546362553260856214285594997,
0.11973530885247993523862602081, 2.67986639580866039430435698841, 2.85648157764595498643303704705, 4.37101309350045477559156752467, 6.07358250193529846721869453871, 7.09514562625993481401660675326, 7.77191970843025353159737549648, 8.522426741335626551069198058567, 9.400338103261589052785905522508, 10.15340663450173151553935558574