| L(s) = 1 | + (6.41 + 2.08i)2-s + (4.20 − 3.05i)3-s + (23.8 + 17.3i)4-s + (−9.80 − 30.1i)5-s + (33.3 − 10.8i)6-s + (−46.3 + 63.8i)7-s + (53.6 + 73.8i)8-s + (8.34 − 25.6i)9-s − 214. i·10-s + (−96.8 + 72.4i)11-s + 153.·12-s + (69.5 + 22.5i)13-s + (−430. + 313. i)14-s + (−133. − 96.9i)15-s + (44.4 + 136. i)16-s + (445. − 144. i)17-s + ⋯ |
| L(s) = 1 | + (1.60 + 0.521i)2-s + (0.467 − 0.339i)3-s + (1.49 + 1.08i)4-s + (−0.392 − 1.20i)5-s + (0.926 − 0.300i)6-s + (−0.946 + 1.30i)7-s + (0.838 + 1.15i)8-s + (0.103 − 0.317i)9-s − 2.14i·10-s + (−0.800 + 0.599i)11-s + 1.06·12-s + (0.411 + 0.133i)13-s + (−2.19 + 1.59i)14-s + (−0.593 − 0.430i)15-s + (0.173 + 0.534i)16-s + (1.54 − 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.06281 + 0.472533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.06281 + 0.472533i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.20 + 3.05i)T \) |
| 11 | \( 1 + (96.8 - 72.4i)T \) |
| good | 2 | \( 1 + (-6.41 - 2.08i)T + (12.9 + 9.40i)T^{2} \) |
| 5 | \( 1 + (9.80 + 30.1i)T + (-505. + 367. i)T^{2} \) |
| 7 | \( 1 + (46.3 - 63.8i)T + (-741. - 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-69.5 - 22.5i)T + (2.31e4 + 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-445. + 144. i)T + (6.75e4 - 4.90e4i)T^{2} \) |
| 19 | \( 1 + (154. + 212. i)T + (-4.02e4 + 1.23e5i)T^{2} \) |
| 23 | \( 1 + 96.1T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-240. + 330. i)T + (-2.18e5 - 6.72e5i)T^{2} \) |
| 31 | \( 1 + (174. - 535. i)T + (-7.47e5 - 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-94.1 - 68.3i)T + (5.79e5 + 1.78e6i)T^{2} \) |
| 41 | \( 1 + (-1.89e3 - 2.61e3i)T + (-8.73e5 + 2.68e6i)T^{2} \) |
| 43 | \( 1 - 729. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (57.5 - 41.8i)T + (1.50e6 - 4.64e6i)T^{2} \) |
| 53 | \( 1 + (69.1 - 212. i)T + (-6.38e6 - 4.63e6i)T^{2} \) |
| 59 | \( 1 + (3.00e3 + 2.18e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (6.17e3 - 2.00e3i)T + (1.12e7 - 8.13e6i)T^{2} \) |
| 67 | \( 1 + 7.10e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-960. - 2.95e3i)T + (-2.05e7 + 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-5.19e3 + 7.15e3i)T + (-8.77e6 - 2.70e7i)T^{2} \) |
| 79 | \( 1 + (890. + 289. i)T + (3.15e7 + 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-872. + 283. i)T + (3.83e7 - 2.78e7i)T^{2} \) |
| 89 | \( 1 - 9.60e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (2.06e3 - 6.36e3i)T + (-7.16e7 - 5.20e7i)T^{2} \) |
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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
−15.79079006489976011437685284141, −14.84894720876435347977031013608, −13.39392308890974809888072179922, −12.56912693084705193727485094459, −12.07160668686971071039693580882, −9.303555122927038115856689056774, −7.82169444595020626183477221989, −6.08550023500405009349801941783, −4.81961266202356973098329040028, −2.95665040868454581236339237393,
3.11959570479260638497320461652, 3.83706415118681654479369493087, 6.01851695002957972917589019952, 7.52750701054018804593027155824, 10.35126517059997142709890038252, 10.80435508636905636537160750621, 12.52792140192142223640470129910, 13.67558587956528010115433839584, 14.35462585076266024753850919266, 15.43147817901462807090797224850
