| L(s) = 1 | + (−0.955 − 1.04i)2-s + (−1.08 + 1.34i)3-s + (−0.173 + 1.99i)4-s + (−2.35 − 0.857i)5-s + (2.44 − 0.153i)6-s + (2.09 + 0.368i)7-s + (2.24 − 1.72i)8-s + (−0.632 − 2.93i)9-s + (1.35 + 3.27i)10-s + (−1.11 − 3.05i)11-s + (−2.49 − 2.40i)12-s + (−2.68 − 3.19i)13-s + (−1.61 − 2.53i)14-s + (3.72 − 2.24i)15-s + (−3.93 − 0.691i)16-s + (−0.433 + 0.250i)17-s + ⋯ |
| L(s) = 1 | + (−0.675 − 0.737i)2-s + (−0.628 + 0.778i)3-s + (−0.0867 + 0.996i)4-s + (−1.05 − 0.383i)5-s + (0.998 − 0.0627i)6-s + (0.790 + 0.139i)7-s + (0.792 − 0.609i)8-s + (−0.210 − 0.977i)9-s + (0.429 + 1.03i)10-s + (−0.335 − 0.921i)11-s + (−0.720 − 0.693i)12-s + (−0.744 − 0.887i)13-s + (−0.431 − 0.676i)14-s + (0.960 − 0.579i)15-s + (−0.984 − 0.172i)16-s + (−0.105 + 0.0606i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.190912 - 0.348277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.190912 - 0.348277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.955 + 1.04i)T \) |
| 3 | \( 1 + (1.08 - 1.34i)T \) |
| good | 5 | \( 1 + (2.35 + 0.857i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-2.09 - 0.368i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.11 + 3.05i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.68 + 3.19i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.433 - 0.250i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.69 + 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.499 + 2.83i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.294 - 0.247i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.22 - 0.216i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (8.56 - 4.94i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.69 + 5.59i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.7 + 3.92i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.561 - 3.18i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + (2.08 - 5.72i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (4.59 + 0.810i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.39 - 7.88i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (2.70 + 4.68i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.58 + 7.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.18 + 10.9i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (8.89 - 10.5i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (12.9 + 7.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.481 - 0.175i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.84791884763132250293283243627, −10.96090945379022684078437588689, −10.35917060341995740312130599930, −9.014789327934130966849307614379, −8.305911065472434395568260430547, −7.26101343742566633977921403473, −5.32523217399878562052238977389, −4.34023003147793288751159637826, −3.07260515438435463839751990231, −0.45521405094496063055077259224,
1.79891137240838147427107772193, 4.46300300487136177345100013766, 5.54588449056811924727681179069, 7.03329986973879990201938204692, 7.45448469310322628098632326499, 8.233396128135044982457639937876, 9.711988140047775549497138724459, 10.81302064143360143649393884441, 11.61343332672838116391075301748, 12.34797905593366510874043855008
