L(s) = 1 | − 0.732·5-s + 3.46·7-s + 6.19·11-s − 2.46·13-s + 0.732·17-s − 4.46·19-s + 8.73·23-s − 4.46·25-s + 2.53·29-s + 2.46·31-s − 2.53·35-s + 2.53·37-s + 9.46·41-s − 5.92·43-s − 6.92·47-s + 4.99·49-s − 7.66·53-s − 4.53·55-s + 1.80·59-s − 1.53·61-s + 1.80·65-s − 3.53·67-s − 6.19·71-s + 11.3·73-s + 21.4·77-s + 13.9·79-s + 12.7·83-s + ⋯ |
L(s) = 1 | − 0.327·5-s + 1.30·7-s + 1.86·11-s − 0.683·13-s + 0.177·17-s − 1.02·19-s + 1.82·23-s − 0.892·25-s + 0.470·29-s + 0.442·31-s − 0.428·35-s + 0.416·37-s + 1.47·41-s − 0.904·43-s − 1.01·47-s + 0.714·49-s − 1.05·53-s − 0.611·55-s + 0.234·59-s − 0.196·61-s + 0.223·65-s − 0.431·67-s − 0.735·71-s + 1.33·73-s + 2.44·77-s + 1.56·79-s + 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.130877280\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.130877280\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 8.73T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 2.46T + 31T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 5.92T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + 3.53T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−9.124704504313158088827163582953, −8.405124705579119643494587238445, −7.69105210399773947620026499265, −6.84603047324666270619531929129, −6.12377124754291391270440856773, −4.86535208042367346346961294623, −4.44066622410118067222608917066, −3.41541263484786576421952861381, −2.07019401611395465061936444601, −1.06713102258613753412883865381,
1.06713102258613753412883865381, 2.07019401611395465061936444601, 3.41541263484786576421952861381, 4.44066622410118067222608917066, 4.86535208042367346346961294623, 6.12377124754291391270440856773, 6.84603047324666270619531929129, 7.69105210399773947620026499265, 8.405124705579119643494587238445, 9.124704504313158088827163582953