| L(s) = 1 | + (−1.21 + 0.717i)2-s + (−1.66 − 0.472i)3-s + (0.970 − 1.74i)4-s + (1.70 + 0.621i)5-s + (2.36 − 0.620i)6-s + (−3.32 − 0.585i)7-s + (0.0715 + 2.82i)8-s + (2.55 + 1.57i)9-s + (−2.52 + 0.467i)10-s + (−0.942 − 2.58i)11-s + (−2.44 + 2.45i)12-s + (−3.98 − 4.74i)13-s + (4.46 − 1.66i)14-s + (−2.55 − 1.84i)15-s + (−2.11 − 3.39i)16-s + (2.90 − 1.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.861 + 0.507i)2-s + (−0.962 − 0.272i)3-s + (0.485 − 0.874i)4-s + (0.763 + 0.277i)5-s + (0.967 − 0.253i)6-s + (−1.25 − 0.221i)7-s + (0.0253 + 0.999i)8-s + (0.851 + 0.524i)9-s + (−0.798 + 0.147i)10-s + (−0.284 − 0.780i)11-s + (−0.705 + 0.708i)12-s + (−1.10 − 1.31i)13-s + (1.19 − 0.445i)14-s + (−0.658 − 0.475i)15-s + (−0.528 − 0.848i)16-s + (0.703 − 0.406i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0114 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0114 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.288551 - 0.285255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.288551 - 0.285255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 - 0.717i)T \) |
| 3 | \( 1 + (1.66 + 0.472i)T \) |
| good | 5 | \( 1 + (-1.70 - 0.621i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.32 + 0.585i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.942 + 2.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.98 + 4.74i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.90 + 1.67i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.396 - 2.25i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (7.06 + 5.92i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.38 + 0.244i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (4.33 - 2.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.54 - 1.84i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.42 + 1.24i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.530 - 3.00i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.184T + 53T^{2} \) |
| 59 | \( 1 + (4.44 - 12.2i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 2.45i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.70 + 3.94i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.83 + 3.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.136 + 0.236i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.710 - 0.846i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.40 + 2.86i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.59 - 3.22i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.5 - 5.29i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.90570385481015440874201800582, −10.82874720268544856823427719078, −9.956524762538017837187154374411, −9.540979732195324758255641060280, −7.77844482602608768321707585045, −6.99512474715698271863009657519, −5.92166993284331358793663637930, −5.35132809253151960136506640321, −2.75253749850637778183759844743, −0.48575733496012861084638845280,
1.88695271381678183793228062360, 3.73788018928897440941604826976, 5.35603589406622978919261578617, 6.55215331337094958499777546418, 7.40261248421099786598128785727, 9.229036791125150803562276683620, 9.741291146535252984964689629523, 10.28073625843738657784296729032, 11.59699731765983689400594931859, 12.53094976959895216851190806949
