L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.23 − 1.21i)3-s + (0.939 − 0.342i)4-s + (1.86 − 1.22i)5-s + (1.42 + 0.977i)6-s + (−1.22 − 0.709i)7-s + (−0.866 + 0.5i)8-s + (0.0679 + 2.99i)9-s + (−1.62 + 1.53i)10-s + (4.15 − 2.40i)11-s + (−1.57 − 0.714i)12-s + (4.38 + 3.68i)13-s + (1.33 + 0.485i)14-s + (−3.80 − 0.739i)15-s + (0.766 − 0.642i)16-s + (−0.869 − 4.92i)17-s + ⋯ |
L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.715 − 0.699i)3-s + (0.469 − 0.171i)4-s + (0.835 − 0.549i)5-s + (0.583 + 0.398i)6-s + (−0.464 − 0.268i)7-s + (−0.306 + 0.176i)8-s + (0.0226 + 0.999i)9-s + (−0.514 + 0.485i)10-s + (1.25 − 0.723i)11-s + (−0.455 − 0.206i)12-s + (1.21 + 1.02i)13-s + (0.356 + 0.129i)14-s + (−0.981 − 0.190i)15-s + (0.191 − 0.160i)16-s + (−0.210 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.782329 - 0.606461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782329 - 0.606461i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.23 + 1.21i)T \) |
| 5 | \( 1 + (-1.86 + 1.22i)T \) |
| 19 | \( 1 + (-3.92 - 1.90i)T \) |
good | 7 | \( 1 + (1.22 + 0.709i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.15 + 2.40i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.38 - 3.68i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.869 + 4.92i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (4.49 - 1.63i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.843 - 4.78i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (8.38 + 4.84i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + (-3.61 + 3.03i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.09 + 8.49i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.195 - 1.10i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.0601 + 0.165i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.697 + 3.95i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.668 + 0.243i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.07 + 11.7i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (11.1 + 4.05i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (0.496 + 0.591i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.74 - 8.03i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.06 - 1.84i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.424 - 0.356i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.40 - 19.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.60865848791208512014150120087, −9.312706754959804574385403370102, −9.147878231016841963910291479431, −7.83306668363657341430950339266, −6.79171165429924633056620557553, −6.18220949574267729474456706346, −5.40444845139389836510269284710, −3.80767077835900723373943328987, −1.91736782219899809968570048246, −0.895906477991350848486098176846,
1.41551650579555078329311989376, 3.10790752953363813327000629115, 4.17060775107373551351679614855, 5.95426547240632511799768948614, 6.05935042221045418289087297269, 7.20983163954897285872412298799, 8.603707788478692251825527725973, 9.487321417748244381826559223559, 9.939510116201404728934238632860, 10.82623645411479991266989236564