| L(s) = 1 | − 3-s + 3.93·5-s − 7-s + 9-s + 2.69·11-s − 1.59·13-s − 3.93·15-s − 1.68·17-s + 2.69·19-s + 21-s − 23-s + 10.4·25-s − 27-s + 4.39·29-s − 8.29·31-s − 2.69·33-s − 3.93·35-s + 6.24·37-s + 1.59·39-s + 2.60·41-s + 10.2·43-s + 3.93·45-s − 3.07·47-s + 49-s + 1.68·51-s + 2.38·53-s + 10.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.75·5-s − 0.377·7-s + 0.333·9-s + 0.813·11-s − 0.441·13-s − 1.01·15-s − 0.407·17-s + 0.619·19-s + 0.218·21-s − 0.208·23-s + 2.09·25-s − 0.192·27-s + 0.816·29-s − 1.49·31-s − 0.469·33-s − 0.664·35-s + 1.02·37-s + 0.254·39-s + 0.406·41-s + 1.56·43-s + 0.586·45-s − 0.448·47-s + 0.142·49-s + 0.235·51-s + 0.328·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.265451654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.265451654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 2.69T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 + 1.68T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 29 | \( 1 - 4.39T + 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 - 1.63T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 + 7.54T + 79T^{2} \) |
| 83 | \( 1 - 2.16T + 83T^{2} \) |
| 89 | \( 1 - 5.84T + 89T^{2} \) |
| 97 | \( 1 - 7.64T + 97T^{2} \) |
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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
−8.767946978304984310245677238912, −7.54064838143291931163676240764, −6.77968947550734771079329170005, −6.19027401220443784716206414830, −5.64231966968148170197105781507, −4.92307937316836030730195512578, −3.95962724483342766351098881158, −2.75838947502701158687289080106, −1.93840447342989340304552813817, −0.932023196189321971144238877995,
0.932023196189321971144238877995, 1.93840447342989340304552813817, 2.75838947502701158687289080106, 3.95962724483342766351098881158, 4.92307937316836030730195512578, 5.64231966968148170197105781507, 6.19027401220443784716206414830, 6.77968947550734771079329170005, 7.54064838143291931163676240764, 8.767946978304984310245677238912
