| L(s) = 1 | + 7.46·2-s + 9.93·3-s + 23.6·4-s − 25·5-s + 74.1·6-s − 61.9·8-s − 144.·9-s − 186.·10-s + 239.·11-s + 235.·12-s − 1.14e3·13-s − 248.·15-s − 1.22e3·16-s − 37.5·17-s − 1.07e3·18-s − 421.·19-s − 592.·20-s + 1.78e3·22-s − 118.·23-s − 615.·24-s + 625·25-s − 8.54e3·26-s − 3.84e3·27-s + 5.65e3·29-s − 1.85e3·30-s − 6.29e3·31-s − 7.12e3·32-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.637·3-s + 0.740·4-s − 0.447·5-s + 0.840·6-s − 0.342·8-s − 0.593·9-s − 0.590·10-s + 0.596·11-s + 0.472·12-s − 1.87·13-s − 0.285·15-s − 1.19·16-s − 0.0315·17-s − 0.783·18-s − 0.267·19-s − 0.331·20-s + 0.787·22-s − 0.0465·23-s − 0.218·24-s + 0.200·25-s − 2.47·26-s − 1.01·27-s + 1.24·29-s − 0.376·30-s − 1.17·31-s − 1.23·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 7.46T + 32T^{2} \) |
| 3 | \( 1 - 9.93T + 243T^{2} \) |
| 11 | \( 1 - 239.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.14e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 37.5T + 1.41e6T^{2} \) |
| 19 | \( 1 + 421.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 118.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.17e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.36e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5T + 8.58e9T^{2} \) |
| 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
−11.13534236325176498413819876093, −9.673898610633366049764586517811, −8.794026030448587573658888743969, −7.63207904966748154390899773689, −6.54537080184821081474831841931, −5.29305653193191695044408794277, −4.36589299576131117123738709535, −3.26955410848621198588392789436, −2.33681814317816808323808570258, 0,
2.33681814317816808323808570258, 3.26955410848621198588392789436, 4.36589299576131117123738709535, 5.29305653193191695044408794277, 6.54537080184821081474831841931, 7.63207904966748154390899773689, 8.794026030448587573658888743969, 9.673898610633366049764586517811, 11.13534236325176498413819876093
