| L(s) = 1 | − 4·5-s + 7-s − 4·11-s + 3·13-s + 7·17-s − 2·19-s − 23-s + 11·25-s − 29-s + 9·31-s − 4·35-s + 2·37-s − 6·41-s − 11·43-s − 6·47-s + 49-s + 9·53-s + 16·55-s − 5·59-s − 6·61-s − 12·65-s − 7·67-s − 7·71-s − 14·73-s − 4·77-s + 6·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.377·7-s − 1.20·11-s + 0.832·13-s + 1.69·17-s − 0.458·19-s − 0.208·23-s + 11/5·25-s − 0.185·29-s + 1.61·31-s − 0.676·35-s + 0.328·37-s − 0.937·41-s − 1.67·43-s − 0.875·47-s + 1/7·49-s + 1.23·53-s + 2.15·55-s − 0.650·59-s − 0.768·61-s − 1.48·65-s − 0.855·67-s − 0.830·71-s − 1.63·73-s − 0.455·77-s + 0.675·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.167664696116707223763778198764, −7.82754494098599337123753060298, −7.08068580428998616978185271748, −6.05495140259777859188099408822, −5.09285010309028921600136805637, −4.41960391934947678645752982589, −3.50570603438794106031082714834, −2.90775362525852210760866139523, −1.28105604099302177257704883763, 0,
1.28105604099302177257704883763, 2.90775362525852210760866139523, 3.50570603438794106031082714834, 4.41960391934947678645752982589, 5.09285010309028921600136805637, 6.05495140259777859188099408822, 7.08068580428998616978185271748, 7.82754494098599337123753060298, 8.167664696116707223763778198764
