L(s) = 1 | + 7·3-s + 5·5-s + 7·7-s + 22·9-s + 9·11-s + 23·13-s + 35·15-s + 41·17-s + 34·19-s + 49·21-s − 6·23-s + 25·25-s − 35·27-s + 131·29-s + 4·31-s + 63·33-s + 35·35-s + 26·37-s + 161·39-s − 260·41-s − 190·43-s + 110·45-s + 167·47-s + 49·49-s + 287·51-s − 368·53-s + 45·55-s + ⋯ |
L(s) = 1 | + 1.34·3-s + 0.447·5-s + 0.377·7-s + 0.814·9-s + 0.246·11-s + 0.490·13-s + 0.602·15-s + 0.584·17-s + 0.410·19-s + 0.509·21-s − 0.0543·23-s + 1/5·25-s − 0.249·27-s + 0.838·29-s + 0.0231·31-s + 0.332·33-s + 0.169·35-s + 0.115·37-s + 0.661·39-s − 0.990·41-s − 0.673·43-s + 0.364·45-s + 0.518·47-s + 1/7·49-s + 0.788·51-s − 0.953·53-s + 0.110·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.331440624\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.331440624\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 23 T + p^{3} T^{2} \) |
| 17 | \( 1 - 41 T + p^{3} T^{2} \) |
| 19 | \( 1 - 34 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 T + p^{3} T^{2} \) |
| 29 | \( 1 - 131 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 T + p^{3} T^{2} \) |
| 37 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 260 T + p^{3} T^{2} \) |
| 43 | \( 1 + 190 T + p^{3} T^{2} \) |
| 47 | \( 1 - 167 T + p^{3} T^{2} \) |
| 53 | \( 1 + 368 T + p^{3} T^{2} \) |
| 59 | \( 1 - 324 T + p^{3} T^{2} \) |
| 61 | \( 1 + 164 T + p^{3} T^{2} \) |
| 67 | \( 1 - 200 T + p^{3} T^{2} \) |
| 71 | \( 1 - 784 T + p^{3} T^{2} \) |
| 73 | \( 1 + 410 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1211 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1132 T + p^{3} T^{2} \) |
| 89 | \( 1 + 72 T + p^{3} T^{2} \) |
| 97 | \( 1 + 707 T + p^{3} T^{2} \) |
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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
−11.41659600191462674991464103431, −10.23110738110269179470316324748, −9.398810084479949208641856549514, −8.527137718400344568353115230964, −7.82021407448414437659586636235, −6.59890538987355797608406616663, −5.26184375719422797650678122604, −3.83367019491121801595879458000, −2.75695599343501972994946863841, −1.46013290372806446505248008030,
1.46013290372806446505248008030, 2.75695599343501972994946863841, 3.83367019491121801595879458000, 5.26184375719422797650678122604, 6.59890538987355797608406616663, 7.82021407448414437659586636235, 8.527137718400344568353115230964, 9.398810084479949208641856549514, 10.23110738110269179470316324748, 11.41659600191462674991464103431