| L(s) = 1 | − 3.43·3-s − 2·4-s + 3.70·5-s + 8.78·9-s + 6.86·12-s − 12.7·15-s + 4·16-s − 7.41·20-s − 5.77·23-s + 8.76·25-s − 19.8·27-s + 1.17·31-s − 17.5·36-s + 2.93·37-s + 32.5·45-s + 2.84·47-s − 13.7·48-s − 7·49-s − 2.01·53-s + 15.3·59-s + 25.4·60-s − 8·64-s + 7.99·67-s + 19.8·69-s + 14.4·71-s − 30.0·75-s + 14.8·80-s + ⋯ |
| L(s) = 1 | − 1.98·3-s − 4-s + 1.65·5-s + 2.92·9-s + 1.98·12-s − 3.28·15-s + 16-s − 1.65·20-s − 1.20·23-s + 1.75·25-s − 3.81·27-s + 0.210·31-s − 2.92·36-s + 0.482·37-s + 4.85·45-s + 0.415·47-s − 1.98·48-s − 49-s − 0.277·53-s + 1.99·59-s + 3.28·60-s − 64-s + 0.977·67-s + 2.38·69-s + 1.72·71-s − 3.47·75-s + 1.65·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8676407877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8676407877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3.43T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 - 2.93T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 2.01T + 53T^{2} \) |
| 59 | \( 1 - 15.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 7.99T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−9.917736414096168101230203468449, −9.219882350300930467009828411716, −7.945395146418489271875933166114, −6.71082809654725712110068847674, −6.10744485385345901520803744035, −5.45240869518680874683894480318, −4.92440542288556650045025389406, −3.95156752957915061229616060420, −1.93929977195890497381908300991, −0.78192839707362352669378575004,
0.78192839707362352669378575004, 1.93929977195890497381908300991, 3.95156752957915061229616060420, 4.92440542288556650045025389406, 5.45240869518680874683894480318, 6.10744485385345901520803744035, 6.71082809654725712110068847674, 7.945395146418489271875933166114, 9.219882350300930467009828411716, 9.917736414096168101230203468449
