| L(s) = 1 | + 3·2-s − 10·3-s + 4-s − 5·5-s − 30·6-s + 22·7-s − 21·8-s + 73·9-s − 15·10-s + 30·11-s − 10·12-s − 46·13-s + 66·14-s + 50·15-s − 71·16-s + 219·18-s + 104·19-s − 5·20-s − 220·21-s + 90·22-s − 42·23-s + 210·24-s + 25·25-s − 138·26-s − 460·27-s + 22·28-s + 66·29-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.92·3-s + 1/8·4-s − 0.447·5-s − 2.04·6-s + 1.18·7-s − 0.928·8-s + 2.70·9-s − 0.474·10-s + 0.822·11-s − 0.240·12-s − 0.981·13-s + 1.25·14-s + 0.860·15-s − 1.10·16-s + 2.86·18-s + 1.25·19-s − 0.0559·20-s − 2.28·21-s + 0.872·22-s − 0.380·23-s + 1.78·24-s + 1/5·25-s − 1.04·26-s − 3.27·27-s + 0.148·28-s + 0.422·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.400146353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400146353\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 10 T + p^{3} T^{2} \) |
| 7 | \( 1 - 22 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 194 T + p^{3} T^{2} \) |
| 37 | \( 1 + 206 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 540 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 432 T + p^{3} T^{2} \) |
| 61 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 848 T + p^{3} T^{2} \) |
| 71 | \( 1 - 174 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 398 T + p^{3} T^{2} \) |
| 83 | \( 1 - 828 T + p^{3} T^{2} \) |
| 89 | \( 1 - 630 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1486 T + p^{3} T^{2} \) |
| 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
−9.407769741541642719239674327203, −8.124252052775068682044452821148, −7.11597246308499139702611720077, −6.54136645624687989490875318744, −5.33403988394224051598615511882, −5.19897564795591272288218420213, −4.40862328787442090367736072782, −3.59769773187800193646474201380, −1.74449510268979532218263542894, −0.56444307676713419799780683725,
0.56444307676713419799780683725, 1.74449510268979532218263542894, 3.59769773187800193646474201380, 4.40862328787442090367736072782, 5.19897564795591272288218420213, 5.33403988394224051598615511882, 6.54136645624687989490875318744, 7.11597246308499139702611720077, 8.124252052775068682044452821148, 9.407769741541642719239674327203
