| L(s) = 1 | − 28.7·3-s − 42.1·7-s + 581.·9-s − 416.·11-s − 966.·13-s + 1.83e3·17-s − 317.·19-s + 1.21e3·21-s + 1.56e3·23-s − 9.72e3·27-s + 7.75e3·29-s − 102.·31-s + 1.19e4·33-s − 1.93e3·37-s + 2.77e4·39-s + 7.99e3·41-s + 1.65e4·43-s + 1.86e4·47-s − 1.50e4·49-s − 5.26e4·51-s + 1.49e4·53-s + 9.12e3·57-s − 1.98e4·59-s − 1.80e4·61-s − 2.45e4·63-s − 5.50e4·67-s − 4.50e4·69-s + ⋯ |
| L(s) = 1 | − 1.84·3-s − 0.325·7-s + 2.39·9-s − 1.03·11-s − 1.58·13-s + 1.53·17-s − 0.201·19-s + 0.598·21-s + 0.618·23-s − 2.56·27-s + 1.71·29-s − 0.0191·31-s + 1.91·33-s − 0.232·37-s + 2.92·39-s + 0.742·41-s + 1.36·43-s + 1.23·47-s − 0.894·49-s − 2.83·51-s + 0.732·53-s + 0.371·57-s − 0.742·59-s − 0.620·61-s − 0.778·63-s − 1.49·67-s − 1.13·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 28.7T + 243T^{2} \) |
| 7 | \( 1 + 42.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 416.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 966.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.83e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 317.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 102.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.99e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.86e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.01e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.07e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.11e4T + 8.58e9T^{2} \) |
| 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
−10.32210149618459398837648146664, −9.496468199196660670474915257222, −7.75966050211441072659133828828, −7.08776763161553775509138154415, −5.98235000067180429959513461917, −5.24352775142049466478840642934, −4.49573515681219835518223467120, −2.74728765563721581162350622261, −0.997471681638018668352888054898, 0,
0.997471681638018668352888054898, 2.74728765563721581162350622261, 4.49573515681219835518223467120, 5.24352775142049466478840642934, 5.98235000067180429959513461917, 7.08776763161553775509138154415, 7.75966050211441072659133828828, 9.496468199196660670474915257222, 10.32210149618459398837648146664
