| L(s) = 1 | − 2·4-s − 2·5-s − 7-s − 11-s + 13-s + 4·16-s − 2·17-s + 4·20-s + 23-s − 25-s + 2·28-s + 8·29-s + 8·31-s + 2·35-s + 7·37-s − 3·41-s + 2·44-s + 2·47-s + 49-s − 2·52-s − 2·53-s + 2·55-s − 4·59-s + 14·61-s − 8·64-s − 2·65-s − 7·67-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 0.377·7-s − 0.301·11-s + 0.277·13-s + 16-s − 0.485·17-s + 0.894·20-s + 0.208·23-s − 1/5·25-s + 0.377·28-s + 1.48·29-s + 1.43·31-s + 0.338·35-s + 1.15·37-s − 0.468·41-s + 0.301·44-s + 0.291·47-s + 1/7·49-s − 0.277·52-s − 0.274·53-s + 0.269·55-s − 0.520·59-s + 1.79·61-s − 64-s − 0.248·65-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.03978492822131, −12.75641119625520, −12.07709052574924, −11.80886594304090, −11.34261558186059, −10.67057435568111, −10.29541152362101, −9.780480247176821, −9.443119519742232, −8.726065395899414, −8.369754687117591, −8.138791750070547, −7.517488094082042, −6.979051035278728, −6.429211143043633, −5.914981618026614, −5.379962408583723, −4.670910124690508, −4.366474991976826, −4.026526626804940, −3.191533872681228, −2.970584722912477, −2.176155276911143, −1.177458571820959, −0.6858955147036268, 0,
0.6858955147036268, 1.177458571820959, 2.176155276911143, 2.970584722912477, 3.191533872681228, 4.026526626804940, 4.366474991976826, 4.670910124690508, 5.379962408583723, 5.914981618026614, 6.429211143043633, 6.979051035278728, 7.517488094082042, 8.138791750070547, 8.369754687117591, 8.726065395899414, 9.443119519742232, 9.780480247176821, 10.29541152362101, 10.67057435568111, 11.34261558186059, 11.80886594304090, 12.07709052574924, 12.75641119625520, 13.03978492822131
