| L(s) = 1 | + (−0.896 + 1.78i)2-s + (2.22 + 2.01i)3-s + (−2.39 − 3.20i)4-s + (−1.17 − 6.65i)5-s + (−5.59 + 2.16i)6-s + (−3.53 − 2.96i)7-s + (7.87 − 1.39i)8-s + (0.867 + 8.95i)9-s + (12.9 + 3.86i)10-s + (0.262 − 1.48i)11-s + (1.15 − 11.9i)12-s + (7.29 − 20.0i)13-s + (8.47 − 3.66i)14-s + (10.8 − 17.1i)15-s + (−4.56 + 15.3i)16-s + (24.0 − 13.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.448 + 0.893i)2-s + (0.740 + 0.672i)3-s + (−0.597 − 0.801i)4-s + (−0.234 − 1.33i)5-s + (−0.932 + 0.360i)6-s + (−0.505 − 0.423i)7-s + (0.984 − 0.174i)8-s + (0.0964 + 0.995i)9-s + (1.29 + 0.386i)10-s + (0.0238 − 0.135i)11-s + (0.0962 − 0.995i)12-s + (0.561 − 1.54i)13-s + (0.605 − 0.261i)14-s + (0.720 − 1.14i)15-s + (−0.285 + 0.958i)16-s + (1.41 − 0.816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28263 - 0.111834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28263 - 0.111834i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.896 - 1.78i)T \) |
| 3 | \( 1 + (-2.22 - 2.01i)T \) |
| good | 5 | \( 1 + (1.17 + 6.65i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.53 + 2.96i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-0.262 + 1.48i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-7.29 + 20.0i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-24.0 + 13.8i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (7.52 + 4.34i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.193 - 0.230i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-34.3 + 12.5i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (15.7 - 13.1i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (51.2 - 29.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-9.84 + 27.0i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (11.8 + 2.09i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-24.8 + 29.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 13.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.1 - 91.6i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (48.6 - 57.9i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-32.0 + 87.9i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (5.14 - 2.96i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (13.7 - 23.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-99.9 + 36.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (80.7 - 29.4i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (15.4 + 8.90i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (3.49 - 19.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.30795624437311299380855265054, −10.51536897138045215580265557304, −9.941233803060834855147181279402, −8.824272858917301258527770035888, −8.303981137314562380307068697955, −7.33997453246881096034489290322, −5.62031644480836574427604767413, −4.80992340583238060736513905718, −3.47402664811958218015687742553, −0.822349208889089511235745298089,
1.77847459393469219852556040623, 3.03286247317033606124507977794, 3.86906477384818784105021410079, 6.31923496079771832975255661755, 7.21495774961321824160241163842, 8.267766022442420412792595936051, 9.232006937327638267176827181430, 10.16319447680086296502940710662, 11.17367987597123956422565559280, 12.13224978572250036459059450802
