| L(s) = 1 | − 25.6·2-s + 146.·4-s − 706.·5-s − 4.56e3·7-s + 9.38e3·8-s + 1.81e4·10-s − 7.14e4·11-s − 2.64e4·13-s + 1.17e5·14-s − 3.15e5·16-s + 3.14e5·17-s − 9.04e5·19-s − 1.03e5·20-s + 1.83e6·22-s + 1.11e5·23-s − 1.45e6·25-s + 6.77e5·26-s − 6.67e5·28-s − 4.38e6·29-s − 9.88e6·31-s + 3.29e6·32-s − 8.07e6·34-s + 3.22e6·35-s − 6.32e6·37-s + 2.31e7·38-s − 6.63e6·40-s − 5.36e6·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.285·4-s − 0.505·5-s − 0.719·7-s + 0.810·8-s + 0.573·10-s − 1.47·11-s − 0.256·13-s + 0.815·14-s − 1.20·16-s + 0.914·17-s − 1.59·19-s − 0.144·20-s + 1.66·22-s + 0.0833·23-s − 0.744·25-s + 0.290·26-s − 0.205·28-s − 1.15·29-s − 1.92·31-s + 0.554·32-s − 1.03·34-s + 0.363·35-s − 0.554·37-s + 1.80·38-s − 0.409·40-s − 0.296·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2363399313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2363399313\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 25.6T + 512T^{2} \) |
| 5 | \( 1 + 706.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.56e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.14e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.64e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.14e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.11e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.38e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.88e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.32e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 5.36e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.24e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.47e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.00e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.58e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.02e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.82e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.94e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.73e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.60e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.74e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.54765928872496892275454189753, −10.95615496250064278965039070892, −10.21020326640196161455487114365, −9.145436146396520928876861215020, −8.018946752032326999034479841257, −7.22871590816500933397902746430, −5.48141470201649627002957583645, −3.85699222767582606127405564420, −2.15713746413767328676582118586, −0.31994093592645233135407272908,
0.31994093592645233135407272908, 2.15713746413767328676582118586, 3.85699222767582606127405564420, 5.48141470201649627002957583645, 7.22871590816500933397902746430, 8.018946752032326999034479841257, 9.145436146396520928876861215020, 10.21020326640196161455487114365, 10.95615496250064278965039070892, 12.54765928872496892275454189753
