| L(s) = 1 | + (−0.939 − 4.42i)2-s + (−1.03 + 5.09i)3-s + (−11.3 + 5.05i)4-s + (9.01 + 6.61i)5-s + (23.4 − 0.231i)6-s + (13.1 − 7.61i)7-s + (11.7 + 16.1i)8-s + (−24.8 − 10.4i)9-s + (20.7 − 46.0i)10-s + (−21.4 + 4.56i)11-s + (−14.0 − 62.9i)12-s + (−13.5 + 63.7i)13-s + (−46.0 − 51.1i)14-s + (−42.9 + 39.1i)15-s + (−6.09 + 6.77i)16-s + (1.96 + 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.332 − 1.56i)2-s + (−0.198 + 0.980i)3-s + (−1.41 + 0.631i)4-s + (0.806 + 0.591i)5-s + (1.59 − 0.0157i)6-s + (0.711 − 0.411i)7-s + (0.518 + 0.713i)8-s + (−0.921 − 0.388i)9-s + (0.656 − 1.45i)10-s + (−0.589 + 0.125i)11-s + (−0.337 − 1.51i)12-s + (−0.288 + 1.35i)13-s + (−0.878 − 0.976i)14-s + (−0.739 + 0.673i)15-s + (−0.0953 + 0.105i)16-s + (0.0279 + 0.0385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.973440 + 0.383650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.973440 + 0.383650i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.03 - 5.09i)T \) |
| 5 | \( 1 + (-9.01 - 6.61i)T \) |
| good | 2 | \( 1 + (0.939 + 4.42i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (-13.1 + 7.61i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (21.4 - 4.56i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (13.5 - 63.7i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 2.69i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-3.72 + 2.70i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (27.9 - 25.1i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (6.82 - 64.9i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (-18.6 - 176. i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (337. - 109. i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-197. - 42.0i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (90.8 - 52.4i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (18.3 + 1.92i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (85.9 - 118. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-601. - 127. i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (83.8 - 17.8i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (417. - 43.8i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (-604. - 439. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-966. - 314. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (20.5 - 195. i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (434. - 975. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (-495. + 1.52e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.51e3 + 159. i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−11.47026684425739302562936923675, −10.89600322376319843767003245917, −10.15118011406442301780509090351, −9.498144340210380763923578323556, −8.539479535561019741370230134401, −6.80920554829289146890785683671, −5.21582570705587615921599142231, −4.14587988634143400642274969141, −2.89822480459523479919574610497, −1.65837498961581347668720370801,
0.48844044048201091522737190865, 2.28596645110730923125225633690, 5.17708970106190552196981602399, 5.54013109847012838494369963872, 6.56685287315820385265368816178, 7.85520936481572234857529470822, 8.193213212027414031865560961249, 9.256851982803248635181636317925, 10.54141624274822216194241349245, 11.93982983352216868790730580302
