| L(s) = 1 | + 3-s − 2.93·5-s + 7-s + 9-s + 2.18·11-s + 0.745·13-s − 2.93·15-s + 0.189·17-s − 4.69·19-s + 21-s − 23-s + 3.61·25-s + 27-s + 3.55·29-s − 3.68·31-s + 2.18·33-s − 2.93·35-s − 0.664·37-s + 0.745·39-s − 7.17·41-s + 1.25·43-s − 2.93·45-s − 5.49·47-s + 49-s + 0.189·51-s − 1.06·53-s − 6.42·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.31·5-s + 0.377·7-s + 0.333·9-s + 0.660·11-s + 0.206·13-s − 0.757·15-s + 0.0459·17-s − 1.07·19-s + 0.218·21-s − 0.208·23-s + 0.723·25-s + 0.192·27-s + 0.659·29-s − 0.661·31-s + 0.381·33-s − 0.496·35-s − 0.109·37-s + 0.119·39-s − 1.12·41-s + 0.191·43-s − 0.437·45-s − 0.801·47-s + 0.142·49-s + 0.0265·51-s − 0.146·53-s − 0.866·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.93T + 5T^{2} \) |
| 11 | \( 1 - 2.18T + 11T^{2} \) |
| 13 | \( 1 - 0.745T + 13T^{2} \) |
| 17 | \( 1 - 0.189T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 + 0.664T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 - 1.25T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + 7.91T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 - 16.8T + 71T^{2} \) |
| 73 | \( 1 + 1.20T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 9.20T + 97T^{2} \) |
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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
−7.67667733572195918036227013610, −6.89231160967650542736141320201, −6.37886776184546405405994371867, −5.25000249133990445431520371047, −4.44881142292906031683589755059, −3.89414259404861715371056479602, −3.32297434628509511644915970312, −2.26853189666214416652210335822, −1.29960953490868288320184326637, 0,
1.29960953490868288320184326637, 2.26853189666214416652210335822, 3.32297434628509511644915970312, 3.89414259404861715371056479602, 4.44881142292906031683589755059, 5.25000249133990445431520371047, 6.37886776184546405405994371867, 6.89231160967650542736141320201, 7.67667733572195918036227013610
