L(s) = 1 | + (0.866 + 0.5i)3-s − 5-s + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + i·23-s + 0.999i·27-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)33-s + (0.866 − 0.5i)35-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s − 5-s + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − 0.999·21-s + i·23-s + 0.999i·27-s + (0.5 − 0.866i)29-s + (−0.5 + 0.866i)33-s + (0.866 − 0.5i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.026854118\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.026854118\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.09925792622545975924037527714, −9.353510067912871408517801243490, −8.980535527016863923149098611023, −7.71219161349956774712772440716, −7.36669278897958561540550653899, −6.17432296555585313667241817814, −4.86188107277528562920809441933, −4.01892293615591860649017238021, −3.25443275792910673354172227913, −2.09468939711511061014624055451,
0.924602959498739877767318876373, 2.89623552428398970464238200300, 3.45722881146812250629576536470, 4.35521183471516629242609588060, 5.94405003289741859636861953826, 6.74337051319561540021444648016, 7.57027325014279353464764764293, 8.356770420349585775398834561233, 8.830259854276491343729171823561, 9.970621913459504505457960143784