| L(s) = 1 | − 3.14·3-s + 1.20·7-s + 6.86·9-s − 3.65·11-s + 0.859·13-s + 6.72·17-s + 1.51·19-s − 3.78·21-s + 23-s − 12.1·27-s + 0.548·29-s + 5.99·31-s + 11.4·33-s + 2.04·37-s − 2.69·39-s − 7.14·41-s + 10.0·43-s + 9.17·47-s − 5.54·49-s − 21.1·51-s + 5.37·53-s − 4.76·57-s + 0.582·59-s − 8.83·61-s + 8.28·63-s + 3.20·67-s − 3.14·69-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 0.456·7-s + 2.28·9-s − 1.10·11-s + 0.238·13-s + 1.63·17-s + 0.348·19-s − 0.826·21-s + 0.208·23-s − 2.33·27-s + 0.101·29-s + 1.07·31-s + 1.99·33-s + 0.335·37-s − 0.432·39-s − 1.11·41-s + 1.53·43-s + 1.33·47-s − 0.792·49-s − 2.95·51-s + 0.738·53-s − 0.631·57-s + 0.0758·59-s − 1.13·61-s + 1.04·63-s + 0.391·67-s − 0.378·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156105772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156105772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 0.859T + 13T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 29 | \( 1 - 0.548T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 + 0.0700T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 89 | \( 1 - 4.96T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
| 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
−7.70986923577935659902213548399, −6.92585855661516401070347685002, −6.19277892919648632811887189173, −5.53887761891344060840709079410, −5.18822482432713081079717442725, −4.52898535566603267760590926172, −3.62359209005685653704552981020, −2.54835359374189664559251565830, −1.31741534659934756902097168715, −0.64867094758168511767369440133,
0.64867094758168511767369440133, 1.31741534659934756902097168715, 2.54835359374189664559251565830, 3.62359209005685653704552981020, 4.52898535566603267760590926172, 5.18822482432713081079717442725, 5.53887761891344060840709079410, 6.19277892919648632811887189173, 6.92585855661516401070347685002, 7.70986923577935659902213548399
