L(s) = 1 | + (−10 + 17.3i)3-s + (37 + 64.0i)5-s + (−78.5 − 135. i)9-s + (−62 + 107. i)11-s + 478·13-s − 1.48e3·15-s + (599 − 1.03e3i)17-s + (−1.52e3 − 2.63e3i)19-s + (−92 − 159. i)23-s + (−1.17e3 + 2.03e3i)25-s − 1.71e3·27-s − 3.28e3·29-s + (2.86e3 − 4.96e3i)31-s + (−1.24e3 − 2.14e3i)33-s + (−5.16e3 − 8.94e3i)37-s + ⋯ |
L(s) = 1 | + (−0.641 + 1.11i)3-s + (0.661 + 1.14i)5-s + (−0.323 − 0.559i)9-s + (−0.154 + 0.267i)11-s + 0.784·13-s − 1.69·15-s + (0.502 − 0.870i)17-s + (−0.967 − 1.67i)19-s + (−0.0362 − 0.0628i)23-s + (−0.376 + 0.651i)25-s − 0.454·27-s − 0.724·29-s + (0.535 − 0.927i)31-s + (−0.198 − 0.343i)33-s + (−0.620 − 1.07i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6363756309\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6363756309\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (10 - 17.3i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (-37 - 64.0i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (62 - 107. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 478T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-599 + 1.03e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.52e3 + 2.63e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (92 + 159. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-2.86e3 + 4.96e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.16e3 + 8.94e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 8.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.18e4 + 2.04e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (5.84e3 - 1.01e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (8.43e3 - 1.46e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-9.24e3 - 1.60e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-7.76e3 + 1.34e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.44e3 + 4.23e3i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.22e4 + 3.85e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 6.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.59e4 + 6.23e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 4.88e4T + 8.58e9T^{2} \) |
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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
−10.40915354379092878042248877332, −9.742208080224699266486639568271, −8.814920390660059116567004800771, −7.30518112164259140487893381265, −6.42987583563835415425886580191, −5.49035294308262908639791900610, −4.52794224933280397978147364901, −3.33674395205035168783289064063, −2.17455745708859994782932089886, −0.16995465515635073577894450080,
1.30409758816197589469217384022, 1.65532519806933513359798353841, 3.65707688917403336997711861529, 5.10952667923917946020897897230, 5.97738547920479228135702661687, 6.54713160563795835363331232449, 8.031998525188269523348244656781, 8.494448480279931156470561034938, 9.746723394328253757376668773537, 10.66783080981340879450835246345