| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 7-s + 4·10-s − 11-s + 2·14-s − 4·16-s + 2·17-s − 6·19-s − 4·20-s + 2·22-s − 2·23-s − 25-s − 2·28-s − 9·29-s + 9·31-s + 8·32-s − 4·34-s + 2·35-s − 4·37-s + 12·38-s + 3·41-s + 2·43-s − 2·44-s + 4·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 0.377·7-s + 1.26·10-s − 0.301·11-s + 0.534·14-s − 16-s + 0.485·17-s − 1.37·19-s − 0.894·20-s + 0.426·22-s − 0.417·23-s − 1/5·25-s − 0.377·28-s − 1.67·29-s + 1.61·31-s + 1.41·32-s − 0.685·34-s + 0.338·35-s − 0.657·37-s + 1.94·38-s + 0.468·41-s + 0.304·43-s − 0.301·44-s + 0.589·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.16523115985062, −13.39603962010486, −13.12109375998334, −12.56132014813204, −11.91202353717571, −11.53191315685906, −11.11852351870093, −10.47146111233321, −10.15240027919551, −9.730822961231397, −9.112709232055321, −8.568442282058814, −8.307007364120580, −7.650013557686934, −7.450415593102137, −6.810456729763252, −6.185295114652303, −5.755960762716467, −4.758241831758007, −4.410914610525175, −3.734029078558786, −3.159638154649198, −2.346431335928552, −1.803155248186331, −1.024057680075686, 0, 0,
1.024057680075686, 1.803155248186331, 2.346431335928552, 3.159638154649198, 3.734029078558786, 4.410914610525175, 4.758241831758007, 5.755960762716467, 6.185295114652303, 6.810456729763252, 7.450415593102137, 7.650013557686934, 8.307007364120580, 8.568442282058814, 9.112709232055321, 9.730822961231397, 10.15240027919551, 10.47146111233321, 11.11852351870093, 11.53191315685906, 11.91202353717571, 12.56132014813204, 13.12109375998334, 13.39603962010486, 14.16523115985062
