| L(s) = 1 | + (1.41 + 2.64i)3-s + 5.65·5-s − 4·7-s + (−5 + 7.48i)9-s + 8.48·11-s + 10.5i·13-s + (8.00 + 14.9i)15-s − 14.9i·17-s − 5.29i·19-s + (−5.65 − 10.5i)21-s − 29.9i·23-s + 7.00·25-s + (−26.8 − 2.64i)27-s − 16.9·29-s + 4·31-s + ⋯ |
| L(s) = 1 | + (0.471 + 0.881i)3-s + 1.13·5-s − 0.571·7-s + (−0.555 + 0.831i)9-s + 0.771·11-s + 0.814i·13-s + (0.533 + 0.997i)15-s − 0.880i·17-s − 0.278i·19-s + (−0.269 − 0.503i)21-s − 1.30i·23-s + 0.280·25-s + (−0.995 − 0.0979i)27-s − 0.585·29-s + 0.129·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51631 + 0.688314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51631 + 0.688314i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - 2.64i)T \) |
| good | 5 | \( 1 - 5.65T + 25T^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 - 8.48T + 121T^{2} \) |
| 13 | \( 1 - 10.5iT - 169T^{2} \) |
| 17 | \( 1 + 14.9iT - 289T^{2} \) |
| 19 | \( 1 + 5.29iT - 361T^{2} \) |
| 23 | \( 1 + 29.9iT - 529T^{2} \) |
| 29 | \( 1 + 16.9T + 841T^{2} \) |
| 31 | \( 1 - 4T + 961T^{2} \) |
| 37 | \( 1 + 52.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 29.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5.29iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 48.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 95.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 89.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 124T + 6.24e3T^{2} \) |
| 83 | \( 1 + 2.82T + 6.88e3T^{2} \) |
| 89 | \( 1 - 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 118T + 9.40e3T^{2} \) |
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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
−14.04785314880408570935374111271, −13.08766582892914806156478698203, −11.61132569964307705971719558370, −10.35110355161242870528211232296, −9.460062704500694166646597056455, −8.841227309134309716470259303149, −6.92294665776047874433245988136, −5.61974447412954744232155675306, −4.13646676666139680601719786919, −2.43590298534683012724796529922,
1.64816861163910483770549558315, 3.34290692331280485161392987416, 5.74772160369392791057882071334, 6.58639289726706434704614253392, 7.982449741174997906631887241189, 9.233323234704780787678602655392, 10.08247895483346465361847334569, 11.66878787542985529239367477440, 12.87531946377046904196763514530, 13.41621228020445400373358747043
