| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.754 + 1.55i)3-s + (0.939 − 0.342i)4-s + (0.486 + 0.408i)5-s + (0.472 − 1.66i)6-s + (−0.719 − 2.54i)7-s + (−0.866 + 0.5i)8-s + (−1.86 − 2.35i)9-s + (−0.550 − 0.317i)10-s + (−3.72 − 4.43i)11-s + (−0.175 + 1.72i)12-s + (−0.505 − 0.0891i)13-s + (1.15 + 2.38i)14-s + (−1.00 + 0.450i)15-s + (0.766 − 0.642i)16-s + (−1.66 + 2.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.435 + 0.900i)3-s + (0.469 − 0.171i)4-s + (0.217 + 0.182i)5-s + (0.192 − 0.680i)6-s + (−0.271 − 0.962i)7-s + (−0.306 + 0.176i)8-s + (−0.620 − 0.784i)9-s + (−0.174 − 0.100i)10-s + (−1.12 − 1.33i)11-s + (−0.0507 + 0.497i)12-s + (−0.140 − 0.0247i)13-s + (0.307 + 0.636i)14-s + (−0.259 + 0.116i)15-s + (0.191 − 0.160i)16-s + (−0.404 + 0.700i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371944 - 0.303126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371944 - 0.303126i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (0.754 - 1.55i)T \) |
| 7 | \( 1 + (0.719 + 2.54i)T \) |
| good | 5 | \( 1 + (-0.486 - 0.408i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (3.72 + 4.43i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.505 + 0.0891i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.66 - 2.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.56 + 2.05i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.488 + 1.34i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (3.07 - 0.541i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.46 + 6.77i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.189 - 0.328i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.877 - 4.97i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.02 + 5.89i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.23 - 1.90i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 3.85iT - 53T^{2} \) |
| 59 | \( 1 + (2.85 + 2.39i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.871 + 2.39i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.36 - 7.75i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.67 + 1.54i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (11.5 - 6.66i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.238 + 1.35i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.33 + 7.57i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (2.08 + 3.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.4 - 13.5i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.82617468953319680411313301284, −10.36697751667541435975968621235, −9.503518079350413865953316510449, −8.500490863483374238548206333549, −7.51769303739113841449399489657, −6.28909055295380542329926117986, −5.50075157378381399467986237065, −4.09379973572236430911510081802, −2.86450613623459247193241140334, −0.40529209964300358093524274848,
1.77180478334755566143978923262, 2.78800519418023625588234628709, 5.07017173196260645317318337862, 5.82351487390796309534553139028, 7.17102479124976934871134849281, 7.62706808666329493327229730405, 8.870177052705696722220081866867, 9.647273728769544778322323624986, 10.65180496454735835462085150114, 11.65860317745547650441396024524
