| L(s) = 1 | + 4·2-s − 3-s + 16·4-s − 4·6-s + 166·7-s + 64·8-s − 242·9-s − 121·11-s − 16·12-s − 692·13-s + 664·14-s + 256·16-s + 738·17-s − 968·18-s + 1.42e3·19-s − 166·21-s − 484·22-s + 1.77e3·23-s − 64·24-s − 2.76e3·26-s + 485·27-s + 2.65e3·28-s − 2.06e3·29-s + 6.24e3·31-s + 1.02e3·32-s + 121·33-s + 2.95e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0641·3-s + 1/2·4-s − 0.0453·6-s + 1.28·7-s + 0.353·8-s − 0.995·9-s − 0.301·11-s − 0.0320·12-s − 1.13·13-s + 0.905·14-s + 1/4·16-s + 0.619·17-s − 0.704·18-s + 0.904·19-s − 0.0821·21-s − 0.213·22-s + 0.701·23-s − 0.0226·24-s − 0.803·26-s + 0.128·27-s + 0.640·28-s − 0.455·29-s + 1.16·31-s + 0.176·32-s + 0.0193·33-s + 0.437·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.731046951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.731046951\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + T + p^{5} T^{2} \) |
| 7 | \( 1 - 166 T + p^{5} T^{2} \) |
| 13 | \( 1 + 692 T + p^{5} T^{2} \) |
| 17 | \( 1 - 738 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1424 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1779 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2064 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6245 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14785 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5304 T + p^{5} T^{2} \) |
| 43 | \( 1 + 17798 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17184 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30726 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34989 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45940 T + p^{5} T^{2} \) |
| 67 | \( 1 + 25343 T + p^{5} T^{2} \) |
| 71 | \( 1 - 13311 T + p^{5} T^{2} \) |
| 73 | \( 1 - 53260 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77234 T + p^{5} T^{2} \) |
| 83 | \( 1 + 55014 T + p^{5} T^{2} \) |
| 89 | \( 1 - 125415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 88807 T + p^{5} T^{2} \) |
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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
−10.18486205571573001144578552174, −9.091048974704884358525046387742, −7.934444779330416257946216950292, −7.47640586989177705551622705709, −6.09115383407124629642224867023, −5.18654590582980070154820346850, −4.64104221004948157056163185344, −3.16377640370239447569317314717, −2.26328043286869344087498674374, −0.873869996874479234352790333521,
0.873869996874479234352790333521, 2.26328043286869344087498674374, 3.16377640370239447569317314717, 4.64104221004948157056163185344, 5.18654590582980070154820346850, 6.09115383407124629642224867023, 7.47640586989177705551622705709, 7.934444779330416257946216950292, 9.091048974704884358525046387742, 10.18486205571573001144578552174
