| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 9-s + 2·10-s + 2·12-s + 4·13-s + 15-s − 4·16-s + 3·17-s + 2·18-s + 7·19-s + 2·20-s + 9·23-s + 25-s + 8·26-s + 27-s + 3·29-s + 2·30-s − 8·32-s + 6·34-s + 2·36-s + 4·37-s + 14·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 1/3·9-s + 0.632·10-s + 0.577·12-s + 1.10·13-s + 0.258·15-s − 16-s + 0.727·17-s + 0.471·18-s + 1.60·19-s + 0.447·20-s + 1.87·23-s + 1/5·25-s + 1.56·26-s + 0.192·27-s + 0.557·29-s + 0.365·30-s − 1.41·32-s + 1.02·34-s + 1/3·36-s + 0.657·37-s + 2.27·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.08381935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.08381935\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.72979413271259, −13.52124067754717, −13.07472425358463, −12.61863490347453, −12.03373056641605, −11.57271042139536, −11.06921643969073, −10.54921698279409, −9.801266179273385, −9.453410823566366, −8.762856629471288, −8.531621157202108, −7.558902278604831, −7.250673954175830, −6.579329457630932, −6.020428313397721, −5.590197984608327, −4.927924477883352, −4.678994639623576, −3.636079255333009, −3.485890520398402, −2.879964430621909, −2.353638614917887, −1.336205546072414, −0.8963003102950892,
0.8963003102950892, 1.336205546072414, 2.353638614917887, 2.879964430621909, 3.485890520398402, 3.636079255333009, 4.678994639623576, 4.927924477883352, 5.590197984608327, 6.020428313397721, 6.579329457630932, 7.250673954175830, 7.558902278604831, 8.531621157202108, 8.762856629471288, 9.453410823566366, 9.801266179273385, 10.54921698279409, 11.06921643969073, 11.57271042139536, 12.03373056641605, 12.61863490347453, 13.07472425358463, 13.52124067754717, 13.72979413271259
