| L(s) = 1 | + 32·2-s + 154.·3-s + 1.02e3·4-s + 9.89e3·5-s + 4.93e3·6-s + 9.23e3·7-s + 3.27e4·8-s − 1.53e5·9-s + 3.16e5·10-s + 1.61e5·11-s + 1.57e5·12-s + 9.36e5·13-s + 2.95e5·14-s + 1.52e6·15-s + 1.04e6·16-s + 4.58e6·17-s − 4.90e6·18-s − 3.20e6·19-s + 1.01e7·20-s + 1.42e6·21-s + 5.15e6·22-s + 2.63e7·23-s + 5.05e6·24-s + 4.90e7·25-s + 2.99e7·26-s − 5.09e7·27-s + 9.45e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.366·3-s + 0.5·4-s + 1.41·5-s + 0.259·6-s + 0.207·7-s + 0.353·8-s − 0.865·9-s + 1.00·10-s + 0.301·11-s + 0.183·12-s + 0.699·13-s + 0.146·14-s + 0.518·15-s + 0.250·16-s + 0.783·17-s − 0.612·18-s − 0.297·19-s + 0.707·20-s + 0.0760·21-s + 0.213·22-s + 0.852·23-s + 0.129·24-s + 1.00·25-s + 0.494·26-s − 0.683·27-s + 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.039591720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039591720\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 11 | \( 1 - 1.61e5T \) |
| good | 3 | \( 1 - 154.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 9.89e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 9.23e3T + 1.97e9T^{2} \) |
| 13 | \( 1 - 9.36e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.58e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 3.20e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.63e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 6.18e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.33e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 3.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.13e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.14e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.38e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 6.66e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.19e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.25e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.47e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 9.83e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.35e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.36e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.07e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.22e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.29e11T + 7.15e21T^{2} \) |
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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
−14.94006547048562962639099661199, −14.00002345905848000307574263328, −13.18782591779744885811482319772, −11.53387610048069171685482231881, −10.02199060381630560030521679838, −8.531219909612427249231720856344, −6.43306841327472552728343219423, −5.26855104893107146035215051321, −3.14081283256949038725614763100, −1.63939093356573842607226200533,
1.63939093356573842607226200533, 3.14081283256949038725614763100, 5.26855104893107146035215051321, 6.43306841327472552728343219423, 8.531219909612427249231720856344, 10.02199060381630560030521679838, 11.53387610048069171685482231881, 13.18782591779744885811482319772, 14.00002345905848000307574263328, 14.94006547048562962639099661199
