| L(s) = 1 | + 6.80·2-s − 489.·3-s − 2.00e3·4-s − 3.09e3·5-s − 3.33e3·6-s − 5.50e4·7-s − 2.75e4·8-s + 6.24e4·9-s − 2.10e4·10-s − 7.25e5·11-s + 9.79e5·12-s + 2.06e6·13-s − 3.74e5·14-s + 1.51e6·15-s + 3.91e6·16-s − 6.85e6·17-s + 4.25e5·18-s + 6.33e6·19-s + 6.18e6·20-s + 2.69e7·21-s − 4.93e6·22-s − 6.43e6·23-s + 1.34e7·24-s − 3.92e7·25-s + 1.40e7·26-s + 5.61e7·27-s + 1.10e8·28-s + ⋯ |
| L(s) = 1 | + 0.150·2-s − 1.16·3-s − 0.977·4-s − 0.442·5-s − 0.174·6-s − 1.23·7-s − 0.297·8-s + 0.352·9-s − 0.0664·10-s − 1.35·11-s + 1.13·12-s + 1.53·13-s − 0.186·14-s + 0.514·15-s + 0.932·16-s − 1.17·17-s + 0.0530·18-s + 0.587·19-s + 0.432·20-s + 1.43·21-s − 0.204·22-s − 0.208·23-s + 0.345·24-s − 0.804·25-s + 0.231·26-s + 0.752·27-s + 1.20·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.2887928841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2887928841\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 6.43e6T \) |
| good | 2 | \( 1 - 6.80T + 2.04e3T^{2} \) |
| 3 | \( 1 + 489.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 3.09e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 5.50e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 7.25e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.06e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.33e6T + 1.16e14T^{2} \) |
| 29 | \( 1 - 1.28e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.29e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.52e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.25e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 3.02e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.51e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.31e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 8.81e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.31e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.31e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 4.25e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.82e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.19e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.05e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.39e10T + 7.15e21T^{2} \) |
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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
−15.63763707159955816003053585837, −13.53081495608931227124813799158, −12.80092468438513823876993180739, −11.32854996551927816318402771160, −10.05178742176668692532421280548, −8.446139978928073976650103082934, −6.39074469216529935644663260573, −5.17987782660886703373246990589, −3.52792555824017801664203452047, −0.37920356752326696246101170198,
0.37920356752326696246101170198, 3.52792555824017801664203452047, 5.17987782660886703373246990589, 6.39074469216529935644663260573, 8.446139978928073976650103082934, 10.05178742176668692532421280548, 11.32854996551927816318402771160, 12.80092468438513823876993180739, 13.53081495608931227124813799158, 15.63763707159955816003053585837
