| L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (4.5 − 2.59i)5-s + (−3.17 + 5.49i)7-s + 2.82i·8-s + 7.34·10-s + (−8.17 − 4.71i)11-s + (−9.84 − 17.0i)13-s + (−7.77 + 4.48i)14-s + (−2.00 + 3.46i)16-s + 1.90i·17-s + 4.69·19-s + (8.99 + 5.19i)20-s + (−6.67 − 11.5i)22-s + (−8.17 + 4.71i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.900 − 0.519i)5-s + (−0.453 + 0.785i)7-s + 0.353i·8-s + 0.734·10-s + (−0.743 − 0.429i)11-s + (−0.757 − 1.31i)13-s + (−0.555 + 0.320i)14-s + (−0.125 + 0.216i)16-s + 0.112i·17-s + 0.247·19-s + (0.449 + 0.259i)20-s + (−0.303 − 0.525i)22-s + (−0.355 + 0.205i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.56546 + 0.350845i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56546 + 0.350845i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-4.5 + 2.59i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.17 - 5.49i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.17 + 4.71i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (9.84 + 17.0i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 1.90iT - 289T^{2} \) |
| 19 | \( 1 - 4.69T + 361T^{2} \) |
| 23 | \( 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-2.84 - 1.64i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-53.5 + 30.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (0.477 - 0.826i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (79.2 - 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 + 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.4 + 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-76.1 - 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.27323221303044369919713619489, −13.95964822875690544166479871671, −12.96331648643574167350711311847, −12.25155142118112179990854765845, −10.48695314411271394991603011313, −9.210350850012109184743499359920, −7.82224578454848618532931742350, −5.99042078936862224261862283907, −5.18292863907704008178311093880, −2.79957378522606009976112233670,
2.43487193110263834237567461591, 4.42673951975014801029203195194, 6.14029241001700672711663948878, 7.32433086988763557207921758102, 9.612939992636769175091269424656, 10.29909810684846432399930405960, 11.64347456722610218820388027934, 12.99934847147306525485195098790, 13.85474101506675231864455963233, 14.66330513244444513122432033662
