| L(s) = 1 | − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 2·13-s + 14-s − 16-s − 17-s + 3·19-s − 2·20-s + 23-s − 25-s + 2·26-s + 28-s − 29-s + 7·31-s − 5·32-s + 34-s − 2·35-s − 3·38-s + 6·40-s + 5·41-s − 6·43-s − 46-s − 11·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.242·17-s + 0.688·19-s − 0.447·20-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s − 0.185·29-s + 1.25·31-s − 0.883·32-s + 0.171·34-s − 0.338·35-s − 0.486·38-s + 0.948·40-s + 0.780·41-s − 0.914·43-s − 0.147·46-s − 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.27938483719600, −12.89511807581979, −12.25165011975208, −11.79538432915626, −11.27707896781450, −10.66599033377273, −10.13379939981072, −9.924752941603219, −9.460329305103947, −9.158300939640133, −8.529186205147309, −8.105213368909359, −7.619353497623598, −7.011545984085714, −6.626365275311666, −5.976204828562723, −5.462783801051535, −5.017551558821482, −4.466291382358210, −3.935958177389776, −3.183931251146194, −2.659846892763831, −1.969400439285055, −1.415575751406481, −0.7444067419362038, 0,
0.7444067419362038, 1.415575751406481, 1.969400439285055, 2.659846892763831, 3.183931251146194, 3.935958177389776, 4.466291382358210, 5.017551558821482, 5.462783801051535, 5.976204828562723, 6.626365275311666, 7.011545984085714, 7.619353497623598, 8.105213368909359, 8.529186205147309, 9.158300939640133, 9.460329305103947, 9.924752941603219, 10.13379939981072, 10.66599033377273, 11.27707896781450, 11.79538432915626, 12.25165011975208, 12.89511807581979, 13.27938483719600
