L(s) = 1 | + (0.355 − 0.615i)5-s + (−2.70 + 1.56i)7-s + (−14.3 + 8.30i)11-s + (9.17 − 15.8i)13-s + 9.69·17-s − 8.20i·19-s + (1.94 + 1.12i)23-s + (12.2 + 21.2i)25-s + (−20.8 − 36.0i)29-s + (21.6 + 12.4i)31-s + 2.21i·35-s + 40.3·37-s + (−25.6 + 44.5i)41-s + (−56.6 + 32.7i)43-s + (29.2 − 16.9i)47-s + ⋯ |
L(s) = 1 | + (0.0710 − 0.123i)5-s + (−0.386 + 0.223i)7-s + (−1.30 + 0.754i)11-s + (0.705 − 1.22i)13-s + 0.570·17-s − 0.431i·19-s + (0.0847 + 0.0489i)23-s + (0.489 + 0.848i)25-s + (−0.717 − 1.24i)29-s + (0.697 + 0.402i)31-s + 0.0634i·35-s + 1.09·37-s + (−0.626 + 1.08i)41-s + (−1.31 + 0.760i)43-s + (0.623 − 0.359i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8973710529\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8973710529\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.355 + 0.615i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (2.70 - 1.56i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (14.3 - 8.30i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.17 + 15.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 9.69T + 289T^{2} \) |
| 19 | \( 1 + 8.20iT - 361T^{2} \) |
| 23 | \( 1 + (-1.94 - 1.12i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (20.8 + 36.0i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-21.6 - 12.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (25.6 - 44.5i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (56.6 - 32.7i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-29.2 + 16.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-66.2 - 38.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.35 + 2.35i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.5 + 19.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 38.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (94.4 - 54.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (113. - 65.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 38.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (12.1 + 21.1i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.659011201481094215858909105051, −8.388394635005777733795262285988, −7.941758286938372356507829738050, −7.08337652635444561554759396285, −6.03841973853265394848441927527, −5.37003641091959898381302553334, −4.55766332930852712360175807529, −3.25642065701719071032944479220, −2.60583208169307327800904039898, −1.15620954533361650884309499831,
0.25147011660944960238556165452, 1.68921806736655414495636179684, 2.91916727644425601528503164308, 3.70521426244560024975736686666, 4.78883320106987142251029215005, 5.70875725145592312100851337223, 6.45716203053301809846974688757, 7.26546356960876731961262439336, 8.203561408691824382890027139779, 8.769005421384103897636790512064