| L(s) = 1 | + 2.74·2-s + 6.45·3-s − 0.486·4-s + 6.42·5-s + 17.6·6-s − 11.8·7-s − 23.2·8-s + 14.6·9-s + 17.6·10-s − 3.13·12-s − 13·13-s − 32.4·14-s + 41.4·15-s − 59.8·16-s − 28.8·17-s + 40.1·18-s + 126.·19-s − 3.12·20-s − 76.4·21-s − 8.23·23-s − 150.·24-s − 83.7·25-s − 35.6·26-s − 79.6·27-s + 5.75·28-s − 174.·29-s + 113.·30-s + ⋯ |
| L(s) = 1 | + 0.969·2-s + 1.24·3-s − 0.0607·4-s + 0.574·5-s + 1.20·6-s − 0.639·7-s − 1.02·8-s + 0.543·9-s + 0.556·10-s − 0.0754·12-s − 0.277·13-s − 0.619·14-s + 0.713·15-s − 0.935·16-s − 0.412·17-s + 0.526·18-s + 1.52·19-s − 0.0349·20-s − 0.794·21-s − 0.0746·23-s − 1.27·24-s − 0.669·25-s − 0.268·26-s − 0.567·27-s + 0.0388·28-s − 1.11·29-s + 0.691·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 2.74T + 8T^{2} \) |
| 3 | \( 1 - 6.45T + 27T^{2} \) |
| 5 | \( 1 - 6.42T + 125T^{2} \) |
| 7 | \( 1 + 11.8T + 343T^{2} \) |
| 17 | \( 1 + 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.23T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 688.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 767.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 791.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 708.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 821.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 44.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 886.T + 9.12e5T^{2} \) |
| 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
−8.971512079162068766949397787434, −7.80668156442400582077849261607, −7.09614734203234607032767785041, −5.89534208327463383929315960870, −5.43797978947645761520733108895, −4.22721306357416553421393132037, −3.44236832381613241077019548984, −2.83330816685276902219861071129, −1.81774730001448680398503944112, 0,
1.81774730001448680398503944112, 2.83330816685276902219861071129, 3.44236832381613241077019548984, 4.22721306357416553421393132037, 5.43797978947645761520733108895, 5.89534208327463383929315960870, 7.09614734203234607032767785041, 7.80668156442400582077849261607, 8.971512079162068766949397787434
