| L(s) = 1 | + 10.7·2-s + 24.8·3-s + 84.1·4-s + 25·5-s + 267.·6-s − 58.6·7-s + 561.·8-s + 375.·9-s + 269.·10-s − 121·11-s + 2.09e3·12-s + 215.·13-s − 631.·14-s + 621.·15-s + 3.36e3·16-s + 1.59e3·17-s + 4.04e3·18-s − 361·19-s + 2.10e3·20-s − 1.45e3·21-s − 1.30e3·22-s − 344.·23-s + 1.39e4·24-s + 625·25-s + 2.32e3·26-s + 3.28e3·27-s − 4.93e3·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.59·3-s + 2.62·4-s + 0.447·5-s + 3.03·6-s − 0.452·7-s + 3.10·8-s + 1.54·9-s + 0.851·10-s − 0.301·11-s + 4.19·12-s + 0.354·13-s − 0.861·14-s + 0.713·15-s + 3.28·16-s + 1.33·17-s + 2.94·18-s − 0.229·19-s + 1.17·20-s − 0.721·21-s − 0.574·22-s − 0.135·23-s + 4.95·24-s + 0.200·25-s + 0.674·26-s + 0.867·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(17.30129581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.30129581\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 10.7T + 32T^{2} \) |
| 3 | \( 1 - 24.8T + 243T^{2} \) |
| 7 | \( 1 + 58.6T + 1.68e4T^{2} \) |
| 13 | \( 1 - 215.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.59e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 344.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 94.4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.65e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 468.T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.99e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.64e4T + 8.58e9T^{2} \) |
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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
−9.274409019089702409210653990515, −7.992802850749585401177897950515, −7.50038557053088698317199072346, −6.41745593359111802165632724190, −5.69430426801379486305243036186, −4.65633270310963215780296458400, −3.64292660281570408149487134894, −3.17337350611580256791052922639, −2.34370090064071849707910648459, −1.47783464239398829414047436837,
1.47783464239398829414047436837, 2.34370090064071849707910648459, 3.17337350611580256791052922639, 3.64292660281570408149487134894, 4.65633270310963215780296458400, 5.69430426801379486305243036186, 6.41745593359111802165632724190, 7.50038557053088698317199072346, 7.992802850749585401177897950515, 9.274409019089702409210653990515
