| L(s) = 1 | + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (156 + 270. i)5-s + (−161.5 + 279. i)7-s + 511.·8-s − 2.49e3·10-s + (1.86e3 − 3.22e3i)11-s + (7.08e3 + 1.22e4i)13-s + (−1.29e3 − 2.23e3i)14-s + (−2.04e3 + 3.54e3i)16-s − 1.59e4·17-s + 2.24e4·19-s + (9.98e3 − 1.72e4i)20-s + (1.48e4 + 2.57e4i)22-s + (2.88e4 + 5.00e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.558 + 0.966i)5-s + (−0.177 + 0.308i)7-s + 0.353·8-s − 0.789·10-s + (0.421 − 0.729i)11-s + (0.894 + 1.55i)13-s + (−0.125 − 0.217i)14-s + (−0.125 + 0.216i)16-s − 0.785·17-s + 0.749·19-s + (0.279 − 0.483i)20-s + (0.297 + 0.516i)22-s + (0.495 + 0.857i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.764040983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764040983\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-156 - 270. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (161.5 - 279. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.86e3 + 3.22e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-7.08e3 - 1.22e4i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-2.88e4 - 5.00e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-8.33e4 + 1.44e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (4.74e4 + 8.21e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.13e5 - 5.43e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.12e4 + 3.67e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (6.17e5 - 1.06e6i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.07e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.23e6 + 2.14e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.43e6 - 2.48e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.50e5 + 1.29e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.73e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.51e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-5.90e5 + 1.02e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (5.58e5 - 9.66e5i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 9.36e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.01e6 + 1.76e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−11.54982270745521962193746299441, −11.00984574633140183309357516436, −9.602918432552615002593038358053, −9.050731658935884637861966736338, −7.69969791124155265410332606926, −6.39884913741107267959060441681, −6.11294517695656758348573004458, −4.32006346023210792978053981346, −2.79753116843316276213976782823, −1.31333493666739910456883580042,
0.58559212633771622085649779422, 1.50676143875448754014717157756, 3.04798904711665549597888295593, 4.45548341869070580586840537060, 5.58249743083386214584894121231, 7.05972311863445637643009704978, 8.410257668707143090069459493226, 9.138972003382514074366718933482, 10.16641145952829371959858174007, 10.98542177571393625142763392530
