| 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
−11.66385906732714241327974927457, −10.65946599302463667078572367818, −10.24258630306180343392027664381, −8.749372380095970495320216727381, −7.71412790951671814241780684456, −6.75897687291350750478289912964, −5.88024026189349650580677265267, −4.43431401951027522239678908317, −1.89842558933629298058980831016, −0.19730800567180010474074154676,
1.89449415976847051607116396320, 3.84700669417669365432456660044, 5.48899014795386747462234950712, 6.53004515365560442659613620785, 7.76006010975116736738267494787, 8.918419870407747117839319259964, 9.850922492485098583198109250195, 10.76611154221436724168573637361, 11.59498225637037515877578415470, 12.25614587992843846792153579876
