| L(s) = 1 | + (−1.90 − 0.617i)2-s + (−2.83 + 0.968i)3-s + (3.23 + 2.34i)4-s + (−0.0648 + 0.0128i)5-s + (5.99 − 0.0887i)6-s + (1.99 + 4.81i)7-s + (−4.70 − 6.46i)8-s + (7.12 − 5.49i)9-s + (0.131 + 0.0154i)10-s + (−0.715 + 1.07i)11-s + (−11.4 − 3.53i)12-s + (0.396 − 1.99i)13-s + (−0.821 − 10.3i)14-s + (0.171 − 0.0994i)15-s + (4.96 + 15.2i)16-s + (−2.86 + 2.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.951 − 0.308i)2-s + (−0.946 + 0.322i)3-s + (0.809 + 0.587i)4-s + (−0.0129 + 0.00257i)5-s + (0.999 − 0.0147i)6-s + (0.284 + 0.687i)7-s + (−0.588 − 0.808i)8-s + (0.791 − 0.610i)9-s + (0.0131 + 0.00154i)10-s + (−0.0650 + 0.0974i)11-s + (−0.955 − 0.294i)12-s + (0.0305 − 0.153i)13-s + (−0.0586 − 0.742i)14-s + (0.0114 − 0.00662i)15-s + (0.310 + 0.950i)16-s + (−0.168 + 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0546930 + 0.226252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0546930 + 0.226252i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.90 + 0.617i)T \) |
| 3 | \( 1 + (2.83 - 0.968i)T \) |
| good | 5 | \( 1 + (0.0648 - 0.0128i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-1.99 - 4.81i)T + (-34.6 + 34.6i)T^{2} \) |
| 11 | \( 1 + (0.715 - 1.07i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (-0.396 + 1.99i)T + (-156. - 64.6i)T^{2} \) |
| 17 | \( 1 + (2.86 - 2.86i)T - 289iT^{2} \) |
| 19 | \( 1 + (26.7 + 5.32i)T + (333. + 138. i)T^{2} \) |
| 23 | \( 1 + (13.8 - 33.4i)T + (-374. - 374. i)T^{2} \) |
| 29 | \( 1 + (14.9 + 22.3i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 - 11.2iT - 961T^{2} \) |
| 37 | \( 1 + (48.4 - 9.63i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-11.6 + 28.1i)T + (-1.18e3 - 1.18e3i)T^{2} \) |
| 43 | \( 1 + (15.6 - 23.4i)T + (-707. - 1.70e3i)T^{2} \) |
| 47 | \( 1 + (9.30 - 9.30i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-16.6 + 24.8i)T + (-1.07e3 - 2.59e3i)T^{2} \) |
| 59 | \( 1 + (5.73 - 1.14i)T + (3.21e3 - 1.33e3i)T^{2} \) |
| 61 | \( 1 + (11.2 + 16.8i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + (21.8 + 32.7i)T + (-1.71e3 + 4.14e3i)T^{2} \) |
| 71 | \( 1 + (61.2 - 25.3i)T + (3.56e3 - 3.56e3i)T^{2} \) |
| 73 | \( 1 + (23.9 - 57.8i)T + (-3.76e3 - 3.76e3i)T^{2} \) |
| 79 | \( 1 + (-89.0 + 89.0i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + (22.0 - 110. i)T + (-6.36e3 - 2.63e3i)T^{2} \) |
| 89 | \( 1 + (-54.3 - 131. i)T + (-5.60e3 + 5.60e3i)T^{2} \) |
| 97 | \( 1 - 138. iT - 9.40e3T^{2} \) |
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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
−12.25614587992843846792153579876, −11.59498225637037515877578415470, −10.76611154221436724168573637361, −9.850922492485098583198109250195, −8.918419870407747117839319259964, −7.76006010975116736738267494787, −6.53004515365560442659613620785, −5.48899014795386747462234950712, −3.84700669417669365432456660044, −1.89449415976847051607116396320,
0.19730800567180010474074154676, 1.89842558933629298058980831016, 4.43431401951027522239678908317, 5.88024026189349650580677265267, 6.75897687291350750478289912964, 7.71412790951671814241780684456, 8.749372380095970495320216727381, 10.24258630306180343392027664381, 10.65946599302463667078572367818, 11.66385906732714241327974927457
