| L(s) = 1 | + 2·5-s − 2·7-s − 3·9-s − 2·11-s − 13-s + 6·17-s − 6·19-s + 8·23-s − 25-s + 2·29-s + 10·31-s − 4·35-s − 6·37-s − 6·41-s + 4·43-s − 6·45-s − 2·47-s − 3·49-s + 6·53-s − 4·55-s − 10·59-s − 2·61-s + 6·63-s − 2·65-s + 10·67-s + 10·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s − 0.277·13-s + 1.45·17-s − 1.37·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.79·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s − 1.30·59-s − 0.256·61-s + 0.755·63-s − 0.248·65-s + 1.22·67-s + 1.18·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8454832086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8454832086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.42487118812446396082482572505, −14.26563333906249084450038791649, −13.28342306795187960857166875332, −12.20226607119849826715844365538, −10.63948592226509211592254466097, −9.648146620493893432528626292635, −8.328372177248017987314125945770, −6.53255603589242536760694775465, −5.33159922228181014436369840982, −2.88461211972819480737788508228,
2.88461211972819480737788508228, 5.33159922228181014436369840982, 6.53255603589242536760694775465, 8.328372177248017987314125945770, 9.648146620493893432528626292635, 10.63948592226509211592254466097, 12.20226607119849826715844365538, 13.28342306795187960857166875332, 14.26563333906249084450038791649, 15.42487118812446396082482572505
