| L(s) = 1 | − 10.8·2-s − 243·3-s − 1.92e3·4-s + 3.57e3·5-s + 2.64e3·6-s − 6.64e4·7-s + 4.33e4·8-s + 5.90e4·9-s − 3.89e4·10-s − 5.86e5·11-s + 4.68e5·12-s − 2.48e5·13-s + 7.24e5·14-s − 8.67e5·15-s + 3.47e6·16-s + 3.17e6·17-s − 6.43e5·18-s − 8.29e6·19-s − 6.88e6·20-s + 1.61e7·21-s + 6.38e6·22-s + 4.26e7·23-s − 1.05e7·24-s − 3.60e7·25-s + 2.71e6·26-s − 1.43e7·27-s + 1.28e8·28-s + ⋯ |
| L(s) = 1 | − 0.240·2-s − 0.577·3-s − 0.941·4-s + 0.510·5-s + 0.139·6-s − 1.49·7-s + 0.467·8-s + 0.333·9-s − 0.123·10-s − 1.09·11-s + 0.543·12-s − 0.185·13-s + 0.360·14-s − 0.295·15-s + 0.829·16-s + 0.542·17-s − 0.0802·18-s − 0.768·19-s − 0.481·20-s + 0.863·21-s + 0.264·22-s + 1.38·23-s − 0.270·24-s − 0.738·25-s + 0.0447·26-s − 0.192·27-s + 1.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 10.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.57e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.64e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 5.86e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.48e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.17e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.29e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.26e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.96e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.28e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.75e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.12e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.03e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.14e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.08e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 8.65e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.80e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 7.45e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.81e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.85e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.72e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.76e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.57e10T + 7.15e21T^{2} \) |
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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
−9.951671406416369390173095578470, −9.496308173073160752095539449925, −8.256058427752288716172477228118, −7.01021554187658508019265453457, −5.89214631431764790727648474624, −5.08409075711198732399461682739, −3.78869663037892841241061932879, −2.55297829826557157886102812349, −0.852007744921931124803239668588, 0,
0.852007744921931124803239668588, 2.55297829826557157886102812349, 3.78869663037892841241061932879, 5.08409075711198732399461682739, 5.89214631431764790727648474624, 7.01021554187658508019265453457, 8.256058427752288716172477228118, 9.496308173073160752095539449925, 9.951671406416369390173095578470
