L(s) = 1 | + (−1.5 − 2.59i)3-s + (2 − 3.46i)5-s + (3 − 1.73i)7-s + (−4.5 + 7.79i)9-s + (−10.5 + 6.06i)11-s + (−11 + 19.0i)13-s − 12·15-s − 11·17-s − 15.5i·19-s + (−9 − 5.19i)21-s + (−21 − 12.1i)23-s + (4.50 + 7.79i)25-s + 27·27-s + (17 + 29.4i)29-s + (−6 − 3.46i)31-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.400 − 0.692i)5-s + (0.428 − 0.247i)7-s + (−0.5 + 0.866i)9-s + (−0.954 + 0.551i)11-s + (−0.846 + 1.46i)13-s − 0.800·15-s − 0.647·17-s − 0.820i·19-s + (−0.428 − 0.247i)21-s + (−0.913 − 0.527i)23-s + (0.180 + 0.311i)25-s + 27-s + (0.586 + 1.01i)29-s + (−0.193 − 0.111i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3527382112\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3527382112\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
good | 5 | \( 1 + (-2 + 3.46i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-3 + 1.73i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (10.5 - 6.06i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (11 - 19.0i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11T + 289T^{2} \) |
| 19 | \( 1 + 15.5iT - 361T^{2} \) |
| 23 | \( 1 + (21 + 12.1i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-17 - 29.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (6 + 3.46i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + (6.5 - 11.2i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (43.5 - 25.1i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (3 - 1.73i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 52T + 2.80e3T^{2} \) |
| 59 | \( 1 + (46.5 + 26.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (8 + 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-100.5 - 58.0i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + (24 - 13.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-30 + 17.3i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-21.5 - 37.2i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.94700388788674152749362507194, −9.878827530601192824325410902550, −8.945896476952015821213181839251, −8.000455938676038075154183991720, −7.11407006396013270394615029892, −6.36876246564831212367085856633, −4.98493452305527111013193022697, −4.67680193322651878762482940399, −2.46291136828076195058768804096, −1.53745773506965180754326548574,
0.13549330151847848861141960238, 2.42981436620678005191324359228, 3.43144005074733749390539866625, 4.81899609213363227104480983324, 5.60190252630921423660167017704, 6.34211221004257398356933048986, 7.74693224643566380852781663556, 8.457355086581041208714180344989, 9.825301324173593974240884548717, 10.24941443540511059274347981873