| L(s) = 1 | − 3-s + 4·5-s + 4·7-s + 9-s − 4·11-s − 4·13-s − 4·15-s + 6·17-s + 19-s − 4·21-s − 6·23-s + 11·25-s − 27-s + 2·29-s + 2·31-s + 4·33-s + 16·35-s + 4·37-s + 4·39-s − 6·41-s + 4·43-s + 4·45-s − 2·47-s + 9·49-s − 6·51-s − 6·53-s − 16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.03·15-s + 1.45·17-s + 0.229·19-s − 0.872·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.696·33-s + 2.70·35-s + 0.657·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.596·45-s − 0.291·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.699224775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.699224775\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.88772622244782751806031486453, −10.12348813188252960046029430976, −9.668450097943077425252758564533, −8.213102607783853972181362930470, −7.47928872381719998670522703456, −6.08417078961404314892413228818, −5.32267004942502611581924636194, −4.80382850580496834483645089906, −2.58336626686734098148346747903, −1.53184510895126931162943066328,
1.53184510895126931162943066328, 2.58336626686734098148346747903, 4.80382850580496834483645089906, 5.32267004942502611581924636194, 6.08417078961404314892413228818, 7.47928872381719998670522703456, 8.213102607783853972181362930470, 9.668450097943077425252758564533, 10.12348813188252960046029430976, 10.88772622244782751806031486453
