L(s) = 1 | + 9.31·2-s + 1.66·3-s + 54.7·4-s + 25·5-s + 15.5·6-s − 125.·7-s + 212.·8-s − 240.·9-s + 232.·10-s − 121·11-s + 91.2·12-s + 837.·13-s − 1.16e3·14-s + 41.6·15-s + 222.·16-s + 760.·17-s − 2.23e3·18-s − 361·19-s + 1.36e3·20-s − 209.·21-s − 1.12e3·22-s + 2.71e3·23-s + 353.·24-s + 625·25-s + 7.80e3·26-s − 805.·27-s − 6.87e3·28-s + ⋯ |
L(s) = 1 | + 1.64·2-s + 0.106·3-s + 1.71·4-s + 0.447·5-s + 0.175·6-s − 0.968·7-s + 1.17·8-s − 0.988·9-s + 0.736·10-s − 0.301·11-s + 0.182·12-s + 1.37·13-s − 1.59·14-s + 0.0477·15-s + 0.217·16-s + 0.637·17-s − 1.62·18-s − 0.229·19-s + 0.765·20-s − 0.103·21-s − 0.496·22-s + 1.07·23-s + 0.125·24-s + 0.200·25-s + 2.26·26-s − 0.212·27-s − 1.65·28-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
\(L(3)\) |
\(\approx\) |
\(6.466748617\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.466748617\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 9.31T + 32T^{2} \) |
| 3 | \( 1 - 1.66T + 243T^{2} \) |
| 7 | \( 1 + 125.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 837.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 760.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.60e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.79e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 7.24e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.58e9T^{2} \) |
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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
−9.093810196091767434749669873923, −8.418416667540054785475328231794, −7.04520711024655733654913838256, −6.28019994813547041314658323102, −5.76223743407900888070795856742, −4.96728986465308889205848599391, −3.75651937664434422840856572170, −3.12611901638546003070961602606, −2.39245059675302670393470141901, −0.841027160904928353758228923224,
0.841027160904928353758228923224, 2.39245059675302670393470141901, 3.12611901638546003070961602606, 3.75651937664434422840856572170, 4.96728986465308889205848599391, 5.76223743407900888070795856742, 6.28019994813547041314658323102, 7.04520711024655733654913838256, 8.418416667540054785475328231794, 9.093810196091767434749669873923