| L(s) = 1 | − 3.68·2-s − 9.05·3-s + 5.57·4-s + 7.08·5-s + 33.3·6-s + 28.1·7-s + 8.92·8-s + 55.0·9-s − 26.1·10-s − 15.3·11-s − 50.5·12-s + 2.51·13-s − 103.·14-s − 64.2·15-s − 77.5·16-s − 202.·18-s + 14.3·19-s + 39.5·20-s − 255.·21-s + 56.3·22-s + 180.·23-s − 80.8·24-s − 74.7·25-s − 9.26·26-s − 254.·27-s + 157.·28-s + 41.2·29-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 1.74·3-s + 0.697·4-s + 0.634·5-s + 2.27·6-s + 1.52·7-s + 0.394·8-s + 2.03·9-s − 0.826·10-s − 0.419·11-s − 1.21·12-s + 0.0536·13-s − 1.98·14-s − 1.10·15-s − 1.21·16-s − 2.65·18-s + 0.173·19-s + 0.442·20-s − 2.65·21-s + 0.546·22-s + 1.63·23-s − 0.687·24-s − 0.597·25-s − 0.0699·26-s − 1.81·27-s + 1.06·28-s + 0.264·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6358660850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6358660850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 3.68T + 8T^{2} \) |
| 3 | \( 1 + 9.05T + 27T^{2} \) |
| 5 | \( 1 - 7.08T + 125T^{2} \) |
| 7 | \( 1 - 28.1T + 343T^{2} \) |
| 11 | \( 1 + 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 2.51T + 2.19e3T^{2} \) |
| 19 | \( 1 - 14.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 180.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 41.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 234.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 326.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 57.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 460.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 392.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 615.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 991.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 98.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 698.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
| 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
−11.14992171679573754247817150614, −10.50835387625628892658832688971, −9.728677706779404837048149698286, −8.521125631379976643014903037158, −7.53968919376169217809685064915, −6.55602655569628172647002928663, −5.27137445812745668473326672679, −4.72183095476753956642594272714, −1.77717183517194332236806717685, −0.792103463451985258247776784243,
0.792103463451985258247776784243, 1.77717183517194332236806717685, 4.72183095476753956642594272714, 5.27137445812745668473326672679, 6.55602655569628172647002928663, 7.53968919376169217809685064915, 8.521125631379976643014903037158, 9.728677706779404837048149698286, 10.50835387625628892658832688971, 11.14992171679573754247817150614
