| L(s) = 1 | + (4 − 6.92i)2-s + (−6 − 10.3i)3-s + (−31.9 − 55.4i)4-s + (105 − 181. i)5-s − 96·6-s − 511.·8-s + (1.02e3 − 1.76e3i)9-s + (−840 − 1.45e3i)10-s + (−546 − 945. i)11-s + (−384. + 665. i)12-s + 1.38e3·13-s − 2.52e3·15-s + (−2.04e3 + 3.54e3i)16-s + (−7.35e3 − 1.27e4i)17-s + (−8.17e3 − 1.41e4i)18-s + (1.99e4 − 3.45e4i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.128 − 0.222i)3-s + (−0.249 − 0.433i)4-s + (0.375 − 0.650i)5-s − 0.181·6-s − 0.353·8-s + (0.467 − 0.809i)9-s + (−0.265 − 0.460i)10-s + (−0.123 − 0.214i)11-s + (−0.0641 + 0.111i)12-s + 0.174·13-s − 0.192·15-s + (−0.125 + 0.216i)16-s + (−0.362 − 0.628i)17-s + (−0.330 − 0.572i)18-s + (0.667 − 1.15i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.109547 + 1.72623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.109547 + 1.72623i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (6 + 10.3i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-105 + 181. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (546 + 945. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 1.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (7.35e3 + 1.27e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.99e4 + 3.45e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.43e4 - 5.95e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 1.02e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (1.13e5 + 1.97e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (8.02e4 - 1.39e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.08e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.36e5 - 4.09e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-7.47e5 - 1.29e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.32e6 + 2.28e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (4.13e5 - 7.16e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.30e4 - 1.09e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (4.90e5 + 8.48e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.78e6 + 3.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 5.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-5.97e6 + 1.03e7i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 8.68e6T + 8.07e13T^{2} \) |
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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
−11.97340846078059885743927971798, −11.17613112240268898093399358682, −9.666950857955463661613049163198, −9.073567673262579264793048932462, −7.35517367024588571213581047758, −5.95128066347874311746543461843, −4.79890139004292548580147213950, −3.36962141997240369866973063370, −1.68860387369152877802863941368, −0.47453598492392218486554615297,
1.99558184457698980467203249021, 3.68551956851046683671876064052, 5.04577285125286818961625433879, 6.22492535132899900744500482783, 7.33856453524129645067889982524, 8.474638385277925932249976451089, 10.01980596631144135900057132887, 10.74376527022764659031765963618, 12.20508231084825589085881690460, 13.25482115771916660129935547350
