| L(s) = 1 | + 3.46·2-s − 7·3-s + 3.99·4-s + 13.8·5-s − 24.2·6-s − 22.5·7-s − 13.8·8-s + 22·9-s + 47.9·10-s − 22.5·11-s − 27.9·12-s − 77.9·14-s − 96.9·15-s − 80·16-s − 27·17-s + 76.2·18-s − 88.3·19-s + 55.4·20-s + 157.·21-s − 77.9·22-s − 57·23-s + 96.9·24-s + 66.9·25-s + 35·27-s − 90.0·28-s − 69·29-s − 336·30-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 1.34·3-s + 0.499·4-s + 1.23·5-s − 1.64·6-s − 1.21·7-s − 0.612·8-s + 0.814·9-s + 1.51·10-s − 0.617·11-s − 0.673·12-s − 1.48·14-s − 1.66·15-s − 1.25·16-s − 0.385·17-s + 0.997·18-s − 1.06·19-s + 0.619·20-s + 1.63·21-s − 0.755·22-s − 0.516·23-s + 0.824·24-s + 0.535·25-s + 0.249·27-s − 0.607·28-s − 0.441·29-s − 2.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 3.46T + 8T^{2} \) |
| 3 | \( 1 + 7T + 27T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 + 22.5T + 343T^{2} \) |
| 11 | \( 1 + 22.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 27T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 39.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 85T + 7.95e4T^{2} \) |
| 47 | \( 1 - 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 426T + 1.48e5T^{2} \) |
| 59 | \( 1 - 19.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 + 164.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 306.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
| 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
−12.19897748690645773708945667614, −10.96516330966958945477635777170, −10.08718045321594415059779411528, −9.057830829693235212687455397755, −6.81711797381145033805839784079, −5.94522954094110184298884003891, −5.58197403699315781284562515063, −4.23025508461146700014839836048, −2.54707575418626506781257122783, 0,
2.54707575418626506781257122783, 4.23025508461146700014839836048, 5.58197403699315781284562515063, 5.94522954094110184298884003891, 6.81711797381145033805839784079, 9.057830829693235212687455397755, 10.08718045321594415059779411528, 10.96516330966958945477635777170, 12.19897748690645773708945667614
