| L(s) = 1 | − 8.36i·5-s + 2.43i·7-s − 2.41·11-s − 20.8i·13-s − 13.1·17-s − 32.7·19-s + 33.8i·23-s − 45.0·25-s + 34.7i·29-s + 33.7i·31-s + 20.4·35-s − 30.8i·37-s + 35.1·41-s − 27.9·43-s − 21.9i·47-s + ⋯ |
| L(s) = 1 | − 1.67i·5-s + 0.348i·7-s − 0.219·11-s − 1.60i·13-s − 0.775·17-s − 1.72·19-s + 1.47i·23-s − 1.80·25-s + 1.19i·29-s + 1.08i·31-s + 0.583·35-s − 0.833i·37-s + 0.857·41-s − 0.649·43-s − 0.467i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1908066616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1908066616\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.36iT - 25T^{2} \) |
| 7 | \( 1 - 2.43iT - 49T^{2} \) |
| 11 | \( 1 + 2.41T + 121T^{2} \) |
| 13 | \( 1 + 20.8iT - 169T^{2} \) |
| 17 | \( 1 + 13.1T + 289T^{2} \) |
| 19 | \( 1 + 32.7T + 361T^{2} \) |
| 23 | \( 1 - 33.8iT - 529T^{2} \) |
| 29 | \( 1 - 34.7iT - 841T^{2} \) |
| 31 | \( 1 - 33.7iT - 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 27.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 26.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 93.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 9.97T + 4.48e3T^{2} \) |
| 71 | \( 1 - 64.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 45.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 82.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 59.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 139.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 150.T + 9.40e3T^{2} \) |
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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
−9.047069593524490087950406034197, −8.685587002275088609426624208104, −8.038727716956919305699441752343, −7.07953755110445920625518250764, −5.81794031420637487538173462696, −5.33598985441360789327734248935, −4.55486116522516938644143753682, −3.55547427291054596281169807964, −2.24131876266110515994302927190, −1.11181626507160384118888392211,
0.05204167244406147679628815725, 2.13000329696054904903456383696, 2.59169869129692186636556756691, 4.04654334363902265866972403471, 4.37931559965788984938190632202, 6.06645522751662718375528860852, 6.59684201239882215182228884204, 7.04345713684504285035055137633, 8.078148125427682305644471621827, 8.877775325354995304333269953478
