| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 9-s − 6·11-s − 2·13-s − 4·15-s + 17-s − 4·21-s − 6·23-s − 25-s + 4·27-s + 10·29-s − 2·31-s + 12·33-s + 4·35-s − 6·37-s + 4·39-s − 6·41-s − 8·43-s + 2·45-s − 3·49-s − 2·51-s + 10·53-s − 12·55-s − 8·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s − 1.03·15-s + 0.242·17-s − 0.872·21-s − 1.25·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 0.359·31-s + 2.08·33-s + 0.676·35-s − 0.986·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 1.61·55-s − 1.04·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.848492428379918370925809981265, −8.479714304195993358732213721787, −7.84551712503875045899997453910, −6.75352187088516304852688044292, −5.82444635947202694666915154156, −5.25042504189511314210648359222, −4.65702909889957787987536766416, −2.86666713007843941469888179421, −1.76553002806865541017271379413, 0,
1.76553002806865541017271379413, 2.86666713007843941469888179421, 4.65702909889957787987536766416, 5.25042504189511314210648359222, 5.82444635947202694666915154156, 6.75352187088516304852688044292, 7.84551712503875045899997453910, 8.479714304195993358732213721787, 9.848492428379918370925809981265
