L(s) = 1 | + 14.9·2-s + 81·3-s − 288.·4-s − 659.·5-s + 1.21e3·6-s − 8.23e3·7-s − 1.19e4·8-s + 6.56e3·9-s − 9.84e3·10-s − 8.61e4·11-s − 2.33e4·12-s + 4.37e4·13-s − 1.22e5·14-s − 5.33e4·15-s − 3.08e4·16-s − 3.93e5·17-s + 9.80e4·18-s + 1.66e5·19-s + 1.90e5·20-s − 6.66e5·21-s − 1.28e6·22-s + 1.77e6·23-s − 9.69e5·24-s − 1.51e6·25-s + 6.54e5·26-s + 5.31e5·27-s + 2.37e6·28-s + ⋯ |
L(s) = 1 | + 0.660·2-s + 0.577·3-s − 0.564·4-s − 0.471·5-s + 0.381·6-s − 1.29·7-s − 1.03·8-s + 0.333·9-s − 0.311·10-s − 1.77·11-s − 0.325·12-s + 0.425·13-s − 0.855·14-s − 0.272·15-s − 0.117·16-s − 1.14·17-s + 0.220·18-s + 0.293·19-s + 0.266·20-s − 0.748·21-s − 1.17·22-s + 1.31·23-s − 0.596·24-s − 0.777·25-s + 0.280·26-s + 0.192·27-s + 0.730·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.110067941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.110067941\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
good | 2 | \( 1 - 14.9T + 512T^{2} \) |
| 5 | \( 1 + 659.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.23e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.61e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.37e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.93e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.66e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.77e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.21e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.76e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.27e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.68e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.06e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.10e6T + 3.29e15T^{2} \) |
| 61 | \( 1 - 3.93e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.07e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.90e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.22e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.21e8T + 7.60e17T^{2} \) |
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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
−11.01133340899773673177993057880, −9.888174918626022866202988389465, −9.000946181094382858353050553449, −8.047054774480507072319408658808, −6.81724072539907511790934537360, −5.57949641772379708823834234897, −4.45644604331332679078500128102, −3.35503295797928115358570291559, −2.63212822110591396833912864996, −0.43195098339386381479702676204,
0.43195098339386381479702676204, 2.63212822110591396833912864996, 3.35503295797928115358570291559, 4.45644604331332679078500128102, 5.57949641772379708823834234897, 6.81724072539907511790934537360, 8.047054774480507072319408658808, 9.000946181094382858353050553449, 9.888174918626022866202988389465, 11.01133340899773673177993057880