| L(s) = 1 | + 3-s − 2·4-s + 9-s + 11-s − 2·12-s + 4·16-s + 2·17-s + 3·19-s − 23-s + 27-s − 6·29-s + 33-s − 2·36-s − 6·37-s + 3·41-s + 6·43-s − 2·44-s − 7·47-s + 4·48-s + 2·51-s + 7·53-s + 3·57-s + 3·59-s − 61-s − 8·64-s + 2·67-s − 4·68-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 16-s + 0.485·17-s + 0.688·19-s − 0.208·23-s + 0.192·27-s − 1.11·29-s + 0.174·33-s − 1/3·36-s − 0.986·37-s + 0.468·41-s + 0.914·43-s − 0.301·44-s − 1.02·47-s + 0.577·48-s + 0.280·51-s + 0.961·53-s + 0.397·57-s + 0.390·59-s − 0.128·61-s − 64-s + 0.244·67-s − 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−14.22919279509186, −13.69846718595380, −13.26646810651772, −12.81967051852676, −12.31345681276057, −11.79710545925040, −11.25477492921764, −10.49905850868796, −10.12950824690129, −9.497273559057764, −9.250052050302188, −8.699251982672032, −8.177883571046661, −7.670279027069994, −7.223969452451868, −6.551948157166117, −5.720631087579928, −5.432466900805262, −4.751922388656127, −4.116929052845177, −3.652868148923383, −3.175940606965331, −2.381044389567145, −1.564392834790849, −0.9389267321909348, 0,
0.9389267321909348, 1.564392834790849, 2.381044389567145, 3.175940606965331, 3.652868148923383, 4.116929052845177, 4.751922388656127, 5.432466900805262, 5.720631087579928, 6.551948157166117, 7.223969452451868, 7.670279027069994, 8.177883571046661, 8.699251982672032, 9.250052050302188, 9.497273559057764, 10.12950824690129, 10.49905850868796, 11.25477492921764, 11.79710545925040, 12.31345681276057, 12.81967051852676, 13.26646810651772, 13.69846718595380, 14.22919279509186
