| L(s) = 1 | − 5·5-s + 2·7-s − 30·11-s − 4·13-s − 90·17-s − 28·19-s − 120·23-s + 25·25-s − 210·29-s − 4·31-s − 10·35-s + 200·37-s − 240·41-s − 136·43-s + 120·47-s − 339·49-s + 30·53-s + 150·55-s + 450·59-s − 166·61-s + 20·65-s + 908·67-s + 1.02e3·71-s − 250·73-s − 60·77-s − 916·79-s + 1.14e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.107·7-s − 0.822·11-s − 0.0853·13-s − 1.28·17-s − 0.338·19-s − 1.08·23-s + 1/5·25-s − 1.34·29-s − 0.0231·31-s − 0.0482·35-s + 0.888·37-s − 0.914·41-s − 0.482·43-s + 0.372·47-s − 0.988·49-s + 0.0777·53-s + 0.367·55-s + 0.992·59-s − 0.348·61-s + 0.0381·65-s + 1.65·67-s + 1.70·71-s − 0.400·73-s − 0.0888·77-s − 1.30·79-s + 1.50·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 90 T + p^{3} T^{2} \) |
| 19 | \( 1 + 28 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 T + p^{3} T^{2} \) |
| 37 | \( 1 - 200 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 136 T + p^{3} T^{2} \) |
| 47 | \( 1 - 120 T + p^{3} T^{2} \) |
| 53 | \( 1 - 30 T + p^{3} T^{2} \) |
| 59 | \( 1 - 450 T + p^{3} T^{2} \) |
| 61 | \( 1 + 166 T + p^{3} T^{2} \) |
| 67 | \( 1 - 908 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 73 | \( 1 + 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 916 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1140 T + p^{3} T^{2} \) |
| 89 | \( 1 - 420 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1538 T + p^{3} T^{2} \) |
| 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
−11.60782409652287410694753260102, −10.83710076038086442226546963270, −9.738806915625469386138092395855, −8.532820805327899383605750124384, −7.65506353160362423634253980405, −6.45430359221420968267917888240, −5.09795352696677703538987716585, −3.87032254310677461324075597038, −2.22944842501812921878295857130, 0,
2.22944842501812921878295857130, 3.87032254310677461324075597038, 5.09795352696677703538987716585, 6.45430359221420968267917888240, 7.65506353160362423634253980405, 8.532820805327899383605750124384, 9.738806915625469386138092395855, 10.83710076038086442226546963270, 11.60782409652287410694753260102
