| L(s) = 1 | − 2·2-s − 0.235·3-s + 4·4-s + 11.0·5-s + 0.470·6-s − 18.5·7-s − 8·8-s − 26.9·9-s − 22.0·10-s + 13.9·11-s − 0.940·12-s + 53.9·13-s + 37.0·14-s − 2.59·15-s + 16·16-s + 8.15·17-s + 53.8·18-s + 44.1·20-s + 4.35·21-s − 27.9·22-s − 122.·23-s + 1.88·24-s − 3.20·25-s − 107.·26-s + 12.6·27-s − 74.1·28-s + 221.·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0452·3-s + 0.5·4-s + 0.987·5-s + 0.0319·6-s − 1.00·7-s − 0.353·8-s − 0.997·9-s − 0.697·10-s + 0.383·11-s − 0.0226·12-s + 1.15·13-s + 0.707·14-s − 0.0446·15-s + 0.250·16-s + 0.116·17-s + 0.705·18-s + 0.493·20-s + 0.0452·21-s − 0.270·22-s − 1.10·23-s + 0.0159·24-s − 0.0256·25-s − 0.814·26-s + 0.0903·27-s − 0.500·28-s + 1.41·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.235T + 27T^{2} \) |
| 5 | \( 1 - 11.0T + 125T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 11 | \( 1 - 13.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.15T + 4.91e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 221.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 46.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 317.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 17.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 44.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 638.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 824.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 839.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 578.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 755.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 339.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 765.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 907.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.68e3T + 9.12e5T^{2} \) |
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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
−9.743676642384403430928261193659, −8.721017642139034153080527650202, −8.194662735412139193561776875774, −6.63845489931312932692413356048, −6.26628314054615596418051371034, −5.41535407146702240070213853950, −3.68542162800762350611448717899, −2.69518622321944007099195907561, −1.45640919526272102081602100616, 0,
1.45640919526272102081602100616, 2.69518622321944007099195907561, 3.68542162800762350611448717899, 5.41535407146702240070213853950, 6.26628314054615596418051371034, 6.63845489931312932692413356048, 8.194662735412139193561776875774, 8.721017642139034153080527650202, 9.743676642384403430928261193659
