| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s − 2·14-s + 16-s + 6·17-s − 4·19-s − 2·28-s + 2·29-s + 8·31-s + 32-s + 6·34-s − 8·37-s − 4·38-s + 6·41-s + 2·43-s + 8·47-s − 3·49-s + 6·53-s − 2·56-s + 2·58-s + 6·59-s − 6·61-s + 8·62-s + 64-s + 2·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.377·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 1.31·37-s − 0.648·38-s + 0.937·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.267·56-s + 0.262·58-s + 0.781·59-s − 0.768·61-s + 1.01·62-s + 1/8·64-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.734139347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734139347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.98119928738481, −13.56691853794339, −13.13139603604806, −12.45913058392634, −12.19493234752253, −11.84740357605813, −11.05926543365378, −10.48030346200364, −10.19589639869261, −9.644121637697088, −8.976929568187128, −8.451648124880700, −7.813261303199400, −7.339192024338332, −6.678174438237843, −6.261407696614141, −5.730478696280514, −5.203182183579537, −4.509562685339575, −3.961862187934322, −3.379900333920899, −2.819965622593006, −2.250567578484959, −1.329700313774226, −0.5819344387966077,
0.5819344387966077, 1.329700313774226, 2.250567578484959, 2.819965622593006, 3.379900333920899, 3.961862187934322, 4.509562685339575, 5.203182183579537, 5.730478696280514, 6.261407696614141, 6.678174438237843, 7.339192024338332, 7.813261303199400, 8.451648124880700, 8.976929568187128, 9.644121637697088, 10.19589639869261, 10.48030346200364, 11.05926543365378, 11.84740357605813, 12.19493234752253, 12.45913058392634, 13.13139603604806, 13.56691853794339, 13.98119928738481
