| L(s) = 1 | + (1.27 + 0.614i)2-s + (−1.05 + 1.37i)3-s + (1.24 + 1.56i)4-s + (0.0452 + 0.343i)5-s + (−2.18 + 1.10i)6-s + (0.358 + 1.33i)7-s + (0.622 + 2.75i)8-s + (−0.794 − 2.89i)9-s + (−0.153 + 0.465i)10-s + (−1.53 + 1.17i)11-s + (−3.46 + 0.0699i)12-s + (1.07 − 1.39i)13-s + (−0.365 + 1.92i)14-s + (−0.520 − 0.298i)15-s + (−0.902 + 3.89i)16-s − 0.574·17-s + ⋯ |
| L(s) = 1 | + (0.900 + 0.434i)2-s + (−0.606 + 0.795i)3-s + (0.622 + 0.782i)4-s + (0.0202 + 0.153i)5-s + (−0.891 + 0.452i)6-s + (0.135 + 0.505i)7-s + (0.220 + 0.975i)8-s + (−0.264 − 0.964i)9-s + (−0.0485 + 0.147i)10-s + (−0.463 + 0.355i)11-s + (−0.999 + 0.0202i)12-s + (0.297 − 0.387i)13-s + (−0.0977 + 0.514i)14-s + (−0.134 − 0.0770i)15-s + (−0.225 + 0.974i)16-s − 0.139·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.964492 + 1.45181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.964492 + 1.45181i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 - 0.614i)T \) |
| 3 | \( 1 + (1.05 - 1.37i)T \) |
| good | 5 | \( 1 + (-0.0452 - 0.343i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.358 - 1.33i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.53 - 1.17i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 1.39i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 + 0.574T + 17T^{2} \) |
| 19 | \( 1 + (1.27 - 0.527i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.80 - 0.751i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.67 + 0.352i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-4.87 + 2.81i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.837 + 2.02i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.97 + 11.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-4.52 - 5.89i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (4.21 + 2.43i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.53 - 8.52i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.62 + 0.740i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.645 + 4.90i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-4.11 + 5.35i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (5.04 + 5.04i)T + 71iT^{2} \) |
| 73 | \( 1 + (-8.79 + 8.79i)T - 73iT^{2} \) |
| 79 | \( 1 + (6.57 - 11.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (11.3 + 1.49i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (8.74 - 8.74i)T - 89iT^{2} \) |
| 97 | \( 1 + (-0.426 + 0.738i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.22484046360934897714313424877, −11.18620947984239857483939110330, −10.57503422967563412511931907316, −9.264230754286792774647467810540, −8.175700728970207156118291445598, −6.91141069340341076339312454155, −5.87978551459288373010815026178, −5.09221145046983977526491339451, −4.04611788867778630861381566864, −2.72172620221738248236528571160,
1.18843640711320801805353920720, 2.77341219605022946689004583119, 4.40683385318448332148873729370, 5.38484164615753952406399132167, 6.45620500907518800581857352394, 7.24611414780299055101088615166, 8.523556835805233031373857353337, 10.05761719361543445933252410227, 11.01931093431271715366354793869, 11.47083534554835678902790046878
