| L(s) = 1 | − 11.3·3-s + 2.35e3·5-s + 4.59e3·7-s − 1.95e4·9-s − 3.29e4·11-s + 2.85e4·13-s − 2.66e4·15-s + 4.29e5·17-s − 8.23e5·19-s − 5.20e4·21-s + 1.28e6·23-s + 3.58e6·25-s + 4.44e5·27-s − 3.75e6·29-s + 9.62e6·31-s + 3.73e5·33-s + 1.08e7·35-s + 1.56e7·37-s − 3.23e5·39-s + 2.86e7·41-s − 2.19e7·43-s − 4.60e7·45-s − 4.95e7·47-s − 1.92e7·49-s − 4.86e6·51-s + 9.82e7·53-s − 7.76e7·55-s + ⋯ |
| L(s) = 1 | − 0.0806·3-s + 1.68·5-s + 0.723·7-s − 0.993·9-s − 0.679·11-s + 0.277·13-s − 0.135·15-s + 1.24·17-s − 1.44·19-s − 0.0583·21-s + 0.953·23-s + 1.83·25-s + 0.160·27-s − 0.985·29-s + 1.87·31-s + 0.0547·33-s + 1.21·35-s + 1.37·37-s − 0.0223·39-s + 1.58·41-s − 0.978·43-s − 1.67·45-s − 1.47·47-s − 0.476·49-s − 0.100·51-s + 1.71·53-s − 1.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.214491646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.214491646\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 11.3T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.35e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.59e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.29e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.29e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.28e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.75e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.56e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.86e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.19e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.82e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.59e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.44e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.61e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.46e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.95e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.36e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.08e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.84e8T + 7.60e17T^{2} \) |
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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
−10.65556466346740543846243359979, −9.847296601837300957330312455446, −8.796891264516418713750447032353, −7.920057568939417187189657195692, −6.35085895074001406082597422723, −5.67065324646705674581508728896, −4.78636658447451188791066434032, −2.92524122774672490615087352229, −2.04660457241438441615986370400, −0.876925618477993061076923479268,
0.876925618477993061076923479268, 2.04660457241438441615986370400, 2.92524122774672490615087352229, 4.78636658447451188791066434032, 5.67065324646705674581508728896, 6.35085895074001406082597422723, 7.920057568939417187189657195692, 8.796891264516418713750447032353, 9.847296601837300957330312455446, 10.65556466346740543846243359979
