| L(s) = 1 | + (1.02 − 0.974i)2-s + (−1.02 − 1.39i)3-s + (0.100 − 1.99i)4-s + (1.18 − 0.631i)5-s + (−2.41 − 0.429i)6-s + (4.72 − 0.940i)7-s + (−1.84 − 2.14i)8-s + (−0.893 + 2.86i)9-s + (0.595 − 1.79i)10-s + (2.00 + 1.64i)11-s + (−2.89 + 1.90i)12-s + (−2.66 − 1.42i)13-s + (3.92 − 5.57i)14-s + (−2.09 − 0.999i)15-s + (−3.97 − 0.402i)16-s + (−0.706 − 0.292i)17-s + ⋯ |
| L(s) = 1 | + (0.724 − 0.689i)2-s + (−0.592 − 0.805i)3-s + (0.0504 − 0.998i)4-s + (0.528 − 0.282i)5-s + (−0.984 − 0.175i)6-s + (1.78 − 0.355i)7-s + (−0.651 − 0.758i)8-s + (−0.297 + 0.954i)9-s + (0.188 − 0.568i)10-s + (0.604 + 0.496i)11-s + (−0.834 + 0.551i)12-s + (−0.738 − 0.394i)13-s + (1.04 − 1.48i)14-s + (−0.540 − 0.258i)15-s + (−0.994 − 0.100i)16-s + (−0.171 − 0.0709i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.946568 - 1.73768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946568 - 1.73768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.02 + 0.974i)T \) |
| 3 | \( 1 + (1.02 + 1.39i)T \) |
| good | 5 | \( 1 + (-1.18 + 0.631i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-4.72 + 0.940i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.00 - 1.64i)T + (2.14 + 10.7i)T^{2} \) |
| 13 | \( 1 + (2.66 + 1.42i)T + (7.22 + 10.8i)T^{2} \) |
| 17 | \( 1 + (0.706 + 0.292i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.347 - 1.14i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (7.23 - 4.83i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.806 - 0.661i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-4.36 - 4.36i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.22 + 10.6i)T + (-30.7 - 20.5i)T^{2} \) |
| 41 | \( 1 + (0.972 - 0.649i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.495 - 5.03i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (-6.98 - 2.89i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (7.86 + 6.45i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-5.66 + 3.02i)T + (32.7 - 49.0i)T^{2} \) |
| 61 | \( 1 + (0.987 - 10.0i)T + (-59.8 - 11.9i)T^{2} \) |
| 67 | \( 1 + (7.67 + 0.756i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-13.5 + 2.70i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (1.05 - 5.28i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-0.541 - 1.30i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (3.23 + 10.6i)T + (-69.0 + 46.1i)T^{2} \) |
| 89 | \( 1 + (0.415 - 0.621i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (11.2 - 11.2i)T - 97iT^{2} \) |
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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
−11.31870975449946093334431692990, −10.45767371119420665169544999852, −9.463470438151898381620682836501, −8.036529582091674205946098080943, −7.18895904957604691391825490721, −5.85051831656102594118740553336, −5.15489919091796708011107083569, −4.22700023910713767180468956873, −2.13785597483578980820654608270, −1.36137801912429231333050324435,
2.36622088067705518926352176465, 4.15711535693678631269929202170, 4.78345612065349487798416392853, 5.79845088211732210196798325888, 6.53744072071371245464384279943, 7.947407227570289724609628103224, 8.711061956384551850735676581908, 9.861073338725080237501670171627, 10.99256594979145570372109405178, 11.77158606047950225345583371581
