| L(s) = 1 | + (−0.463 + 1.94i)2-s + (0.0358 + 2.99i)3-s + (−3.56 − 1.80i)4-s + (1.24 + 7.06i)5-s + (−5.85 − 1.32i)6-s + (−7.11 − 5.96i)7-s + (5.16 − 6.10i)8-s + (−8.99 + 0.214i)9-s + (−14.3 − 0.851i)10-s + (0.0360 − 0.204i)11-s + (5.28 − 10.7i)12-s + (−3.20 + 8.80i)13-s + (14.9 − 11.0i)14-s + (−21.1 + 3.98i)15-s + (9.48 + 12.8i)16-s + (6.74 − 3.89i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 + 0.972i)2-s + (0.0119 + 0.999i)3-s + (−0.892 − 0.451i)4-s + (0.249 + 1.41i)5-s + (−0.975 − 0.220i)6-s + (−1.01 − 0.852i)7-s + (0.645 − 0.763i)8-s + (−0.999 + 0.0238i)9-s + (−1.43 − 0.0851i)10-s + (0.00327 − 0.0185i)11-s + (0.440 − 0.897i)12-s + (−0.246 + 0.676i)13-s + (1.06 − 0.790i)14-s + (−1.40 + 0.265i)15-s + (0.593 + 0.805i)16-s + (0.397 − 0.229i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.340495 - 0.500067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.340495 - 0.500067i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.463 - 1.94i)T \) |
| 3 | \( 1 + (-0.0358 - 2.99i)T \) |
| good | 5 | \( 1 + (-1.24 - 7.06i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (7.11 + 5.96i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-0.0360 + 0.204i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.20 - 8.80i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-6.74 + 3.89i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (4.10 + 2.36i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-7.22 - 8.61i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (47.3 - 17.2i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-31.8 + 26.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (38.3 - 22.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-3.37 + 9.28i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-15.6 - 2.76i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (44.0 - 52.5i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 59.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (9.30 + 52.7i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (77.7 - 92.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (28.1 - 77.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (54.9 - 31.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (69.0 - 119. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (83.2 - 30.2i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-77.3 + 28.1i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-152. - 88.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-20.5 + 116. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.20385195129241886264250100214, −11.41362315127352656501867900811, −10.37687677215798751643929346542, −9.906031195413524999259411767398, −9.027540798287646537464260990062, −7.46518763353871791660982358218, −6.71291346773591690293346333976, −5.74624460252936712834303066768, −4.22615793365395506757474215963, −3.16779824548737679513855800191,
0.35492894591172214691544109114, 1.85882814214913222850226503323, 3.22909196428918893437132402457, 5.05885870763659662932046853126, 6.01368708177381955001992123169, 7.71316606141440662469082425770, 8.737958834056314144750310393824, 9.241809361543088930884793474341, 10.42374865326749213753299478361, 11.92536230743829435729863858477
