| L(s) = 1 | + (−6.40 − 3.69i)2-s + (−45.1 − 12.3i)3-s + (−36.6 − 63.4i)4-s + (101. − 175. i)5-s + (243. + 245. i)6-s + (−682. + 598. i)7-s + 1.48e3i·8-s + (1.88e3 + 1.11e3i)9-s + (−1.30e3 + 751. i)10-s + (1.68e3 − 973. i)11-s + (870. + 3.31e3i)12-s + 7.87e3i·13-s + (6.58e3 − 1.30e3i)14-s + (−6.74e3 + 6.68e3i)15-s + (819. − 1.41e3i)16-s + (−4.61e3 − 7.99e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.566 − 0.326i)2-s + (−0.964 − 0.263i)3-s + (−0.286 − 0.495i)4-s + (0.363 − 0.629i)5-s + (0.460 + 0.464i)6-s + (−0.751 + 0.659i)7-s + 1.02i·8-s + (0.861 + 0.508i)9-s + (−0.411 + 0.237i)10-s + (0.381 − 0.220i)11-s + (0.145 + 0.553i)12-s + 0.994i·13-s + (0.641 − 0.127i)14-s + (−0.516 + 0.511i)15-s + (0.0499 − 0.0866i)16-s + (−0.227 − 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.227290 + 0.186367i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227290 + 0.186367i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (45.1 + 12.3i)T \) |
| 7 | \( 1 + (682. - 598. i)T \) |
| good | 2 | \( 1 + (6.40 + 3.69i)T + (64 + 110. i)T^{2} \) |
| 5 | \( 1 + (-101. + 175. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-1.68e3 + 973. i)T + (9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 7.87e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (4.61e3 + 7.99e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-9.01e3 - 5.20e3i)T + (4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (7.09e4 + 4.09e4i)T + (1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.95e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (8.44e4 - 4.87e4i)T + (1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.85e5 - 4.94e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.56e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.83e5 - 6.63e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.37e5 + 7.92e4i)T + (5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.28e6 + 2.21e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.49e6 + 8.64e5i)T + (1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.10e6 - 1.90e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 5.29e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-2.20e6 + 1.27e6i)T + (5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.72e5 - 6.44e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 5.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.08e6 + 8.80e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 3.36e6iT - 8.07e13T^{2} \) |
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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
−16.96995687183403843092714251984, −16.02056007908826701727872823997, −14.08831400823755144571695253373, −12.63629508471614883026488240042, −11.44255419921738341053971099553, −9.941367367702573778516722224428, −8.875015921719965777839807311225, −6.41135578183679581748842591689, −5.06078936003878714849750754587, −1.53045308619339773371196073186,
0.23244411332935152570801042664, 3.83828620182845780880395521262, 6.18259975975654602558282939111, 7.47036619003858301168109617953, 9.595905718073306614439917402454, 10.52162475547025895350675892182, 12.28234743593282407361000776639, 13.51528748368100922267994747875, 15.46942235498899527069073410305, 16.55351503735536592170775976065
