| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s + 2·11-s + 2·13-s − 2·15-s − 17-s − 2·19-s − 2·21-s − 2·23-s − 4·25-s − 4·27-s + 3·29-s − 6·31-s + 4·33-s + 35-s + 4·39-s − 9·41-s + 8·43-s − 45-s + 6·47-s + 49-s − 2·51-s − 14·53-s − 2·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.436·21-s − 0.417·23-s − 4/5·25-s − 0.769·27-s + 0.557·29-s − 1.07·31-s + 0.696·33-s + 0.169·35-s + 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.280·51-s − 1.92·53-s − 0.269·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287201698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287201698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.98517114572129, −13.78497671603972, −13.13989451827769, −12.45349617623707, −12.30030563524695, −11.49112343923376, −11.00901827178120, −10.64222489676969, −9.710396127132862, −9.468777099947494, −8.995954922602928, −8.400301487702961, −7.992006167300076, −7.630377099193144, −6.737069356687052, −6.501456053977409, −5.755646426277198, −5.126976183354323, −4.257245668524163, −3.771557593253042, −3.510036209795931, −2.722776787983063, −2.108699849089473, −1.501803371347609, −0.4441665231591945,
0.4441665231591945, 1.501803371347609, 2.108699849089473, 2.722776787983063, 3.510036209795931, 3.771557593253042, 4.257245668524163, 5.126976183354323, 5.755646426277198, 6.501456053977409, 6.737069356687052, 7.630377099193144, 7.992006167300076, 8.400301487702961, 8.995954922602928, 9.468777099947494, 9.710396127132862, 10.64222489676969, 11.00901827178120, 11.49112343923376, 12.30030563524695, 12.45349617623707, 13.13989451827769, 13.78497671603972, 13.98517114572129
