| L(s) = 1 | + 5·5-s − 22·7-s + 9·11-s + 17·13-s + 75·17-s − 4·19-s − 183·23-s + 25·25-s − 129·29-s − 187·31-s − 110·35-s − 34·37-s − 264·41-s + 443·43-s − 609·47-s + 141·49-s + 228·53-s + 45·55-s − 60·59-s − 454·61-s + 85·65-s − 244·67-s − 444·71-s + 398·73-s − 198·77-s − 349·79-s − 1.03e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.18·7-s + 0.246·11-s + 0.362·13-s + 1.07·17-s − 0.0482·19-s − 1.65·23-s + 1/5·25-s − 0.826·29-s − 1.08·31-s − 0.531·35-s − 0.151·37-s − 1.00·41-s + 1.57·43-s − 1.89·47-s + 0.411·49-s + 0.590·53-s + 0.110·55-s − 0.132·59-s − 0.952·61-s + 0.162·65-s − 0.444·67-s − 0.742·71-s + 0.638·73-s − 0.293·77-s − 0.497·79-s − 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 - 17 T + p^{3} T^{2} \) |
| 17 | \( 1 - 75 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 + 129 T + p^{3} T^{2} \) |
| 31 | \( 1 + 187 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 264 T + p^{3} T^{2} \) |
| 43 | \( 1 - 443 T + p^{3} T^{2} \) |
| 47 | \( 1 + 609 T + p^{3} T^{2} \) |
| 53 | \( 1 - 228 T + p^{3} T^{2} \) |
| 59 | \( 1 + 60 T + p^{3} T^{2} \) |
| 61 | \( 1 + 454 T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 444 T + p^{3} T^{2} \) |
| 73 | \( 1 - 398 T + p^{3} T^{2} \) |
| 79 | \( 1 + 349 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 89 | \( 1 + 852 T + p^{3} T^{2} \) |
| 97 | \( 1 - 914 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.848574306254088756834546258408, −9.330565356972477930997669022149, −8.206673622128876383368728565273, −7.16445544815383182245490957336, −6.17455688130664656771873512598, −5.55077084950083117482668011670, −4.00169962641131954287396349961, −3.10766569528995328481695372530, −1.66864707386110424710874855782, 0,
1.66864707386110424710874855782, 3.10766569528995328481695372530, 4.00169962641131954287396349961, 5.55077084950083117482668011670, 6.17455688130664656771873512598, 7.16445544815383182245490957336, 8.206673622128876383368728565273, 9.330565356972477930997669022149, 9.848574306254088756834546258408
