L(s) = 1 | − 5-s + 5.12·7-s + 4·11-s − 0.561·13-s + 3.12·17-s + 4·19-s − 23-s + 25-s + 8.56·29-s + 1.43·31-s − 5.12·35-s − 7.12·37-s − 0.561·41-s − 9.12·43-s + 3.68·47-s + 19.2·49-s + 4.24·53-s − 4·55-s + 6.24·59-s + 11.1·61-s + 0.561·65-s + 6.24·67-s − 3.68·71-s + 16.5·73-s + 20.4·77-s − 10.2·79-s − 12·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.93·7-s + 1.20·11-s − 0.155·13-s + 0.757·17-s + 0.917·19-s − 0.208·23-s + 0.200·25-s + 1.58·29-s + 0.258·31-s − 0.865·35-s − 1.17·37-s − 0.0876·41-s − 1.39·43-s + 0.537·47-s + 2.74·49-s + 0.583·53-s − 0.539·55-s + 0.813·59-s + 1.42·61-s + 0.0696·65-s + 0.763·67-s − 0.437·71-s + 1.93·73-s + 2.33·77-s − 1.15·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.071876018\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.071876018\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.923769643146157676178375968490, −7.14489581671411361885464710153, −6.62634641992374752768862936009, −5.42761181959295018180435065252, −5.11083247686522509221237981106, −4.24755609493233515197054795986, −3.68936845469457780060311823362, −2.60999853986396129144298821915, −1.53459503631254466209190248204, −0.991002153158277674679225593400,
0.991002153158277674679225593400, 1.53459503631254466209190248204, 2.60999853986396129144298821915, 3.68936845469457780060311823362, 4.24755609493233515197054795986, 5.11083247686522509221237981106, 5.42761181959295018180435065252, 6.62634641992374752768862936009, 7.14489581671411361885464710153, 7.923769643146157676178375968490