| L(s) = 1 | − 2-s − 4-s − 2·5-s − 7-s + 3·8-s + 2·10-s + 14-s − 16-s − 2·17-s + 2·19-s + 2·20-s − 6·23-s − 25-s + 28-s − 29-s − 8·31-s − 5·32-s + 2·34-s + 2·35-s − 2·37-s − 2·38-s − 6·40-s − 6·41-s + 2·43-s + 6·46-s + 4·47-s + 49-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 + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-s + 0.188·28-s − 0.185·29-s − 1.43·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s − 0.328·37-s − 0.324·38-s − 0.948·40-s − 0.937·41-s + 0.304·43-s + 0.884·46-s + 0.583·47-s + 1/7·49-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 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.34332468806024, −12.98102737610331, −12.50642117983074, −11.94251494280501, −11.57947230772937, −11.01696377169722, −10.62058735361776, −10.00975628928269, −9.712929131099748, −9.215330563769849, −8.590958854420952, −8.438487175618650, −7.744008152172732, −7.390432756607890, −7.050716063711897, −6.284801248566633, −5.731222601078472, −5.190877428309781, −4.607241467862126, −4.006232721759323, −3.734937598553192, −3.168738817019156, −2.239712241974906, −1.733066184110343, −0.9541520325499032, 0, 0,
0.9541520325499032, 1.733066184110343, 2.239712241974906, 3.168738817019156, 3.734937598553192, 4.006232721759323, 4.607241467862126, 5.190877428309781, 5.731222601078472, 6.284801248566633, 7.050716063711897, 7.390432756607890, 7.744008152172732, 8.438487175618650, 8.590958854420952, 9.215330563769849, 9.712929131099748, 10.00975628928269, 10.62058735361776, 11.01696377169722, 11.57947230772937, 11.94251494280501, 12.50642117983074, 12.98102737610331, 13.34332468806024
