| L(s) = 1 | − 1.61·2-s − 3·3-s − 5.39·4-s − 1.20·5-s + 4.84·6-s − 28.2·7-s + 21.6·8-s + 9·9-s + 1.95·10-s + 31.7·11-s + 16.1·12-s + 45.6·14-s + 3.62·15-s + 8.22·16-s − 16.0·17-s − 14.5·18-s + 58.5·19-s + 6.51·20-s + 84.8·21-s − 51.3·22-s + 152.·23-s − 64.8·24-s − 123.·25-s − 27·27-s + 152.·28-s + 265.·29-s − 5.85·30-s + ⋯ |
| L(s) = 1 | − 0.570·2-s − 0.577·3-s − 0.674·4-s − 0.108·5-s + 0.329·6-s − 1.52·7-s + 0.955·8-s + 0.333·9-s + 0.0617·10-s + 0.871·11-s + 0.389·12-s + 0.871·14-s + 0.0624·15-s + 0.128·16-s − 0.229·17-s − 0.190·18-s + 0.706·19-s + 0.0728·20-s + 0.881·21-s − 0.497·22-s + 1.38·23-s − 0.551·24-s − 0.988·25-s − 0.192·27-s + 1.02·28-s + 1.69·29-s − 0.0356·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 8T^{2} \) |
| 5 | \( 1 + 1.20T + 125T^{2} \) |
| 7 | \( 1 + 28.2T + 343T^{2} \) |
| 11 | \( 1 - 31.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 16.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 444.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 513.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 619.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 332.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 679.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 88.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 154.T + 9.12e5T^{2} \) |
| 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
−9.912062421977743386293948624314, −9.311603935183450308558580879059, −8.507882246613303790932740741412, −7.17904871867117995307513125351, −6.52407700878589197171245947476, −5.37553446399318453743341837353, −4.23475313713779905448805132441, −3.18866417664960496003954187627, −1.14224213893913145083834393812, 0,
1.14224213893913145083834393812, 3.18866417664960496003954187627, 4.23475313713779905448805132441, 5.37553446399318453743341837353, 6.52407700878589197171245947476, 7.17904871867117995307513125351, 8.507882246613303790932740741412, 9.311603935183450308558580879059, 9.912062421977743386293948624314
