| L(s) = 1 | + 2·5-s − 2·11-s + 13-s − 4·19-s − 4·23-s − 25-s − 8·29-s − 8·31-s + 2·37-s + 6·41-s + 4·43-s − 6·47-s − 7·49-s − 4·53-s − 4·55-s − 6·59-s + 2·61-s + 2·65-s − 4·67-s − 6·71-s − 2·73-s + 16·79-s − 2·83-s + 10·89-s − 8·95-s − 2·97-s + 8·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s + 0.277·13-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.48·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.875·47-s − 49-s − 0.549·53-s − 0.539·55-s − 0.781·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.712·71-s − 0.234·73-s + 1.80·79-s − 0.219·83-s + 1.05·89-s − 0.820·95-s − 0.203·97-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.020748013875403215145758538367, −7.56053238185936013444063524663, −6.47698921732997129090655451070, −5.91637964103366996271590449874, −5.29221798521384458230947646919, −4.30663590846803283794664608933, −3.44321044095513580438308022926, −2.30106247028669018489386209681, −1.68168645810658685804508758078, 0,
1.68168645810658685804508758078, 2.30106247028669018489386209681, 3.44321044095513580438308022926, 4.30663590846803283794664608933, 5.29221798521384458230947646919, 5.91637964103366996271590449874, 6.47698921732997129090655451070, 7.56053238185936013444063524663, 8.020748013875403215145758538367
