| L(s) = 1 | − 3-s + 9-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s − 8·23-s − 27-s − 6·29-s − 8·31-s − 4·33-s − 2·37-s − 2·39-s − 2·41-s + 12·43-s + 8·47-s − 2·51-s + 6·53-s − 4·57-s − 4·59-s − 2·61-s − 12·67-s + 8·69-s − 8·71-s − 14·73-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s − 0.312·41-s + 1.82·43-s + 1.16·47-s − 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 1.46·67-s + 0.963·69-s − 0.949·71-s − 1.63·73-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.07172316827922, −12.57139108479175, −12.08818701994457, −11.80768572825816, −11.37032296431672, −10.88778872741416, −10.31441018481651, −10.02539363758416, −9.256675861974527, −9.070373125626221, −8.623120429871092, −7.673748533122235, −7.473150242414116, −7.127446120421459, −6.180170031430056, −5.991545464281972, −5.669349438149480, −4.968397252275693, −4.307227717230204, −3.745458615660269, −3.623835744842966, −2.706053619551277, −1.888462952769267, −1.477898349225012, −0.8176526652503496, 0,
0.8176526652503496, 1.477898349225012, 1.888462952769267, 2.706053619551277, 3.623835744842966, 3.745458615660269, 4.307227717230204, 4.968397252275693, 5.669349438149480, 5.991545464281972, 6.180170031430056, 7.127446120421459, 7.473150242414116, 7.673748533122235, 8.623120429871092, 9.070373125626221, 9.256675861974527, 10.02539363758416, 10.31441018481651, 10.88778872741416, 11.37032296431672, 11.80768572825816, 12.08818701994457, 12.57139108479175, 13.07172316827922
