| L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s + 13-s + 16-s + 7·17-s + 19-s + 5·22-s + 6·23-s − 26-s + 6·29-s + 31-s − 32-s − 7·34-s − 38-s + 2·41-s − 2·43-s − 5·44-s − 6·46-s + 52-s + 6·53-s − 6·58-s − 4·59-s + 3·61-s − 62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s + 0.277·13-s + 1/4·16-s + 1.69·17-s + 0.229·19-s + 1.06·22-s + 1.25·23-s − 0.196·26-s + 1.11·29-s + 0.179·31-s − 0.176·32-s − 1.20·34-s − 0.162·38-s + 0.312·41-s − 0.304·43-s − 0.753·44-s − 0.884·46-s + 0.138·52-s + 0.824·53-s − 0.787·58-s − 0.520·59-s + 0.384·61-s − 0.127·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927380337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927380337\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.56946438278538, −12.30671823547130, −11.81244196801865, −11.13253510750949, −10.91609273088177, −10.28369629483638, −10.02786098460549, −9.654853814628888, −8.951890432431571, −8.506670312898967, −8.145613036554240, −7.541608016271778, −7.385975053984799, −6.713078648567423, −6.155129572160128, −5.543527877192824, −5.261288882553669, −4.720195321124310, −3.994487579059379, −3.186058742915408, −2.968031865791417, −2.442032653591449, −1.600436540945722, −1.016637604207568, −0.4858105921268497,
0.4858105921268497, 1.016637604207568, 1.600436540945722, 2.442032653591449, 2.968031865791417, 3.186058742915408, 3.994487579059379, 4.720195321124310, 5.261288882553669, 5.543527877192824, 6.155129572160128, 6.713078648567423, 7.385975053984799, 7.541608016271778, 8.145613036554240, 8.506670312898967, 8.951890432431571, 9.654853814628888, 10.02786098460549, 10.28369629483638, 10.91609273088177, 11.13253510750949, 11.81244196801865, 12.30671823547130, 12.56946438278538
