| L(s) = 1 | + (−2 − 3.46i)2-s + (4.18 + 7.25i)3-s + (−7.99 + 13.8i)4-s + (−12.5 − 21.8i)5-s + (16.7 − 29.0i)6-s − 187.·7-s + 63.9·8-s + (86.4 − 149. i)9-s + (−50.3 + 87.2i)10-s − 637.·11-s − 133.·12-s + (−405. + 702. i)13-s + (374. + 648. i)14-s + (105. − 182. i)15-s + (−128 − 221. i)16-s + (238. + 413. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.268 + 0.465i)3-s + (−0.249 + 0.433i)4-s + (−0.225 − 0.390i)5-s + (0.189 − 0.328i)6-s − 1.44·7-s + 0.353·8-s + (0.355 − 0.616i)9-s + (−0.159 + 0.275i)10-s − 1.58·11-s − 0.268·12-s + (−0.665 + 1.15i)13-s + (0.510 + 0.883i)14-s + (0.120 − 0.209i)15-s + (−0.125 − 0.216i)16-s + (0.200 + 0.347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0137469 + 0.0967226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0137469 + 0.0967226i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 19 | \( 1 + (1.56e3 + 109. i)T \) |
| good | 3 | \( 1 + (-4.18 - 7.25i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (12.5 + 21.8i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + 187.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 637.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (405. - 702. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-238. - 413. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-2.22e3 + 3.85e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.32e3 - 4.02e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + 574.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-6.35e3 - 1.10e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (9.15e3 + 1.58e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.00e4 - 1.74e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-5.03e3 + 8.72e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (8.42e3 + 1.45e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (8.27e3 - 1.43e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.06e3 - 3.57e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.12e4 - 3.67e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (2.07e4 + 3.59e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.62e4 + 2.81e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.42e4 - 2.47e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (3.86e4 + 6.69e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−14.76312687352273109884593164071, −12.93752496023407285632443672482, −12.49407092457627928698690381961, −10.62380323995255897695564338744, −9.686957001482125206007144225080, −8.621494327805633241768668042260, −6.77127110438002292878148950124, −4.45274732644581155009062911261, −2.83844289051676168791350855178, −0.05672968219792874059887880713,
2.86251520343501329665304122996, 5.43811846347210717075131258446, 7.10933251318090652908324584520, 7.921185079581426675352563990095, 9.706327134301868993230608239739, 10.68639386627366044645011877653, 12.91693998691514556130192355625, 13.28898436544199481452847018463, 15.11111232458502259050787258127, 15.79398656685934214663374586045
