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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.453672878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.453672878\) |
\(L(2)\) |
|
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
−10.27165398807788491984409335397, −9.537571365893261613431012496338, −8.643354927183824382999635130704, −7.73167091302730197980844438070, −6.23061342126333778188295764995, −5.66499498881111209574205260136, −4.74805766146823919955312296275, −3.94507163453060378997514197424, −1.75528046857123585260037572633, −0.65167973796904829623747616459,
1.57918615951967715087402001431, 2.62032945273377487952441496879, 4.23501929811159741105813942263, 5.51092851598189189259447845587, 6.22018678636770597188758872871, 6.96843315127779173680941484975, 7.78394358430430553374834093274, 9.200493804562168228816905442838, 10.08159043918489944870553414148, 10.88465038368547759776340693703