| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s − 13-s + 15-s + 5·17-s − 21-s − 23-s − 4·25-s − 5·27-s + 29-s − 35-s − 2·37-s − 39-s − 5·41-s + 43-s − 2·45-s − 3·47-s + 49-s + 5·51-s + 9·53-s + 59-s + 5·61-s + 2·63-s − 65-s + 13·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.277·13-s + 0.258·15-s + 1.21·17-s − 0.218·21-s − 0.208·23-s − 4/5·25-s − 0.962·27-s + 0.185·29-s − 0.169·35-s − 0.328·37-s − 0.160·39-s − 0.780·41-s + 0.152·43-s − 0.298·45-s − 0.437·47-s + 1/7·49-s + 0.700·51-s + 1.23·53-s + 0.130·59-s + 0.640·61-s + 0.251·63-s − 0.124·65-s + 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−15.86005073557772, −15.31547990271910, −14.70842270711674, −14.29947573124445, −13.72922519066547, −13.42800275001553, −12.61420268861113, −12.14700719469553, −11.58912102033480, −10.95363359806807, −10.15613490208429, −9.781393520098106, −9.334980737218672, −8.530116378631146, −8.154801967140758, −7.512048790130411, −6.801886243067849, −6.146951195871794, −5.470549871306077, −5.110697312788162, −3.899023116772926, −3.536585360406854, −2.677810030250709, −2.168344098793782, −1.167409421167919, 0,
1.167409421167919, 2.168344098793782, 2.677810030250709, 3.536585360406854, 3.899023116772926, 5.110697312788162, 5.470549871306077, 6.146951195871794, 6.801886243067849, 7.512048790130411, 8.154801967140758, 8.530116378631146, 9.334980737218672, 9.781393520098106, 10.15613490208429, 10.95363359806807, 11.58912102033480, 12.14700719469553, 12.61420268861113, 13.42800275001553, 13.72922519066547, 14.29947573124445, 14.70842270711674, 15.31547990271910, 15.86005073557772
