| L(s) = 1 | + (−26.2 − 45.3i)3-s + (−99.6 + 172. i)5-s + (−279. + 484. i)9-s + (−609. − 1.05e3i)11-s + 6.44e3·13-s + 1.04e4·15-s + (−1.33e4 − 2.31e4i)17-s + (−7.45e3 + 1.29e4i)19-s + (3.67e4 − 6.36e4i)23-s + (1.92e4 + 3.32e4i)25-s − 8.52e4·27-s − 1.06e5·29-s + (−3.34e4 − 5.78e4i)31-s + (−3.19e4 + 5.53e4i)33-s + (2.39e5 − 4.15e5i)37-s + ⋯ |
| L(s) = 1 | + (−0.560 − 0.970i)3-s + (−0.356 + 0.617i)5-s + (−0.127 + 0.221i)9-s + (−0.138 − 0.239i)11-s + 0.814·13-s + 0.798·15-s + (−0.661 − 1.14i)17-s + (−0.249 + 0.431i)19-s + (0.629 − 1.09i)23-s + (0.246 + 0.426i)25-s − 0.834·27-s − 0.810·29-s + (−0.201 − 0.348i)31-s + (−0.154 + 0.267i)33-s + (0.778 − 1.34i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1233907454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1233907454\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (26.2 + 45.3i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (99.6 - 172. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (609. + 1.05e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 6.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (1.33e4 + 2.31e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (7.45e3 - 1.29e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.67e4 + 6.36e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (3.34e4 + 5.78e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.39e5 + 4.15e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 6.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (5.05e5 - 8.75e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.71e4 + 2.96e4i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-2.21e5 - 3.83e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-5.70e5 + 9.87e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.15e6 - 3.74e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.83e6 - 3.18e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.27e6 - 7.40e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 1.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.78e6 - 4.81e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.72e7T + 8.07e13T^{2} \) |
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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
−11.28214006784498022783142891386, −11.07294294818559203321474041388, −9.537114904490032922706764408022, −8.323767138146035863083960902712, −7.18306516783960989989576724160, −6.59345680504772517635660239053, −5.47860384921091361982561338109, −3.90616793167776405555196925372, −2.52949531173661521046451577216, −1.08345390428256736115630164645,
0.03769574961545899779633908542, 1.62113719499228249659975478702, 3.56223904798319913416939771650, 4.51914883626294864979106170958, 5.37504250134008535731227265315, 6.61064088078392944903217079373, 8.079929649069061966367007538786, 8.969947741269293092522390473441, 10.03898789114404177613881807107, 10.92583458591536093198052998292
