L(s) = 1 | + 3-s + 5-s − 2·7-s + 9-s − 2·11-s + 15-s + 17-s − 2·21-s − 8·23-s + 25-s + 27-s + 6·29-s + 3·31-s − 2·33-s − 2·35-s − 4·37-s − 4·41-s + 12·43-s + 45-s + 47-s − 3·49-s + 51-s − 5·53-s − 2·55-s − 10·59-s + 14·61-s − 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s + 0.242·17-s − 0.436·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.538·31-s − 0.348·33-s − 0.338·35-s − 0.657·37-s − 0.624·41-s + 1.82·43-s + 0.149·45-s + 0.145·47-s − 3/7·49-s + 0.140·51-s − 0.686·53-s − 0.269·55-s − 1.30·59-s + 1.79·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.89030427796576, −15.45946374001804, −14.54546138411824, −14.20012512124395, −13.68041575495846, −13.22460976942629, −12.56672369895881, −12.20309992106618, −11.54518762300909, −10.52250665248324, −10.37151501441923, −9.697739974296118, −9.280754452941357, −8.546018278918171, −7.987566172532374, −7.519768000252144, −6.594058915787967, −6.287992651525685, −5.523976600350866, −4.840170190445986, −4.051527288367558, −3.409383206171221, −2.668064744375432, −2.156607130646852, −1.147977681299296, 0,
1.147977681299296, 2.156607130646852, 2.668064744375432, 3.409383206171221, 4.051527288367558, 4.840170190445986, 5.523976600350866, 6.287992651525685, 6.594058915787967, 7.519768000252144, 7.987566172532374, 8.546018278918171, 9.280754452941357, 9.697739974296118, 10.37151501441923, 10.52250665248324, 11.54518762300909, 12.20309992106618, 12.56672369895881, 13.22460976942629, 13.68041575495846, 14.20012512124395, 14.54546138411824, 15.45946374001804, 15.89030427796576