| L(s) = 1 | − 3-s + 5-s − 7-s − 2·9-s − 2·11-s + 6·13-s − 15-s − 5·17-s + 21-s + 3·23-s + 25-s + 5·27-s − 4·29-s − 2·31-s + 2·33-s − 35-s − 37-s − 6·39-s − 11·41-s + 43-s − 2·45-s + 49-s + 5·51-s + 3·53-s − 2·55-s − 2·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.603·11-s + 1.66·13-s − 0.258·15-s − 1.21·17-s + 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.742·29-s − 0.359·31-s + 0.348·33-s − 0.169·35-s − 0.164·37-s − 0.960·39-s − 1.71·41-s + 0.152·43-s − 0.298·45-s + 1/7·49-s + 0.700·51-s + 0.412·53-s − 0.269·55-s − 0.260·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.30018645402559, −12.86612339607294, −12.47918357139244, −11.63887000345330, −11.42418399141949, −10.98865836080529, −10.47188198563001, −10.26976748416134, −9.398829461988186, −9.000116556434243, −8.603781524099896, −8.253597088099679, −7.448988205447348, −6.927148288111761, −6.417175712747035, −6.054012496360145, −5.585811008890943, −5.130390599591385, −4.533942858773481, −3.896672473037425, −3.218680576695995, −2.895717480744703, −2.032034033411266, −1.546084311094715, −0.6715113342204400, 0,
0.6715113342204400, 1.546084311094715, 2.032034033411266, 2.895717480744703, 3.218680576695995, 3.896672473037425, 4.533942858773481, 5.130390599591385, 5.585811008890943, 6.054012496360145, 6.417175712747035, 6.927148288111761, 7.448988205447348, 8.253597088099679, 8.603781524099896, 9.000116556434243, 9.398829461988186, 10.26976748416134, 10.47188198563001, 10.98865836080529, 11.42418399141949, 11.63887000345330, 12.47918357139244, 12.86612339607294, 13.30018645402559
