| L(s) = 1 | + (2.76 + 1.16i)3-s + (4.03 + 6.98i)5-s + (−3.90 − 2.25i)7-s + (6.29 + 6.42i)9-s + (−3.25 − 1.88i)11-s + (3.52 + 6.10i)13-s + (3.03 + 23.9i)15-s + 0.517·17-s + 16.4i·19-s + (−8.17 − 10.7i)21-s + (27.7 − 15.9i)23-s + (−19.9 + 34.6i)25-s + (9.94 + 25.1i)27-s + (−9.48 + 16.4i)29-s + (−13.1 + 7.58i)31-s + ⋯ |
| L(s) = 1 | + (0.921 + 0.387i)3-s + (0.806 + 1.39i)5-s + (−0.557 − 0.321i)7-s + (0.699 + 0.714i)9-s + (−0.296 − 0.171i)11-s + (0.271 + 0.469i)13-s + (0.202 + 1.59i)15-s + 0.0304·17-s + 0.864i·19-s + (−0.389 − 0.512i)21-s + (1.20 − 0.695i)23-s + (−0.799 + 1.38i)25-s + (0.368 + 0.929i)27-s + (−0.327 + 0.566i)29-s + (−0.423 + 0.244i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0769 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0769 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.607736119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607736119\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.76 - 1.16i)T \) |
| good | 5 | \( 1 + (-4.03 - 6.98i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3.90 + 2.25i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (3.25 + 1.88i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-3.52 - 6.10i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 0.517T + 289T^{2} \) |
| 19 | \( 1 - 16.4iT - 361T^{2} \) |
| 23 | \( 1 + (-27.7 + 15.9i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (9.48 - 16.4i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (13.1 - 7.58i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 0.592T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-12.3 - 21.4i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (27.8 + 16.0i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (52.4 + 30.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 0.664T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-30.5 + 17.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.7 - 58.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-74.4 + 42.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 56.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 131.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-126. - 73.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (87.1 + 50.2i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 25.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + (48.2 - 83.5i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.53751091740423373931998370040, −9.933763579007219651291726635844, −9.180100376088687170270833007068, −8.130979385700577282485096121735, −7.02858466929409853765640869337, −6.48670790871322697644911546077, −5.18085131430339663018774797713, −3.68841143243678148518478546609, −3.01230801530503545568877671266, −1.90958024041492050794293070951,
0.904968801258671753898831611780, 2.13953012796228603392015858149, 3.31180402628096819011366998673, 4.71204661479453372981727852767, 5.62634203034764381106971476687, 6.69610860897105706678899961840, 7.82664749900605213788402601175, 8.663071721829665513295893257331, 9.379542570043352989803014428856, 9.780353361800730548504671076691
