| L(s) = 1 | − 1.60·2-s + 5.64·3-s − 5.41·4-s + 18.5·5-s − 9.08·6-s + 5.73·7-s + 21.5·8-s + 4.90·9-s − 29.8·10-s − 29.7·11-s − 30.5·12-s − 12.6·13-s − 9.22·14-s + 104.·15-s + 8.58·16-s − 106.·17-s − 7.89·18-s + 20.5·19-s − 100.·20-s + 32.3·21-s + 47.8·22-s − 157.·23-s + 121.·24-s + 220.·25-s + 20.3·26-s − 124.·27-s − 31.0·28-s + ⋯ |
| L(s) = 1 | − 0.568·2-s + 1.08·3-s − 0.676·4-s + 1.66·5-s − 0.618·6-s + 0.309·7-s + 0.953·8-s + 0.181·9-s − 0.945·10-s − 0.815·11-s − 0.735·12-s − 0.270·13-s − 0.176·14-s + 1.80·15-s + 0.134·16-s − 1.51·17-s − 0.103·18-s + 0.248·19-s − 1.12·20-s + 0.336·21-s + 0.464·22-s − 1.43·23-s + 1.03·24-s + 1.76·25-s + 0.153·26-s − 0.889·27-s − 0.209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 31 | \( 1 \) |
| good | 2 | \( 1 + 1.60T + 8T^{2} \) |
| 3 | \( 1 - 5.64T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 - 5.73T + 343T^{2} \) |
| 11 | \( 1 + 29.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 239.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 87.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 513.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 261.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 761.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 687.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 390.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 845.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 391.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
| 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
−9.260529772888385309137039546325, −8.520779645425943260114533910722, −7.952864203582581232548479273558, −6.82453933741223809040403873414, −5.62586091406580100320637872950, −4.92774640863577932620364362741, −3.65811766280388286219784685705, −2.25540911494470576374578964415, −1.83879586756413126722153354020, 0,
1.83879586756413126722153354020, 2.25540911494470576374578964415, 3.65811766280388286219784685705, 4.92774640863577932620364362741, 5.62586091406580100320637872950, 6.82453933741223809040403873414, 7.952864203582581232548479273558, 8.520779645425943260114533910722, 9.260529772888385309137039546325
