| L(s) = 1 | + (4.32 + 0.763i)5-s + (−2.73 + 2.29i)7-s + (14.4 − 2.54i)11-s + (−3.67 + 1.33i)13-s + (5.96 + 3.44i)17-s + (14.1 + 24.4i)19-s + (−0.832 + 0.992i)23-s + (−5.33 − 1.94i)25-s + (12.9 − 35.7i)29-s + (41.7 + 35.0i)31-s + (−13.6 + 7.85i)35-s + (18.5 − 32.1i)37-s + (14.0 + 38.6i)41-s + (−0.615 − 3.49i)43-s + (27.5 + 32.7i)47-s + ⋯ |
| L(s) = 1 | + (0.865 + 0.152i)5-s + (−0.391 + 0.328i)7-s + (1.31 − 0.231i)11-s + (−0.282 + 0.102i)13-s + (0.350 + 0.202i)17-s + (0.744 + 1.28i)19-s + (−0.0361 + 0.0431i)23-s + (−0.213 − 0.0777i)25-s + (0.448 − 1.23i)29-s + (1.34 + 1.12i)31-s + (−0.388 + 0.224i)35-s + (0.500 − 0.867i)37-s + (0.343 + 0.943i)41-s + (−0.0143 − 0.0812i)43-s + (0.585 + 0.697i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.91182 + 0.416190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91182 + 0.416190i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-4.32 - 0.763i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (2.73 - 2.29i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-14.4 + 2.54i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.67 - 1.33i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-5.96 - 3.44i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.1 - 24.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.832 - 0.992i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-12.9 + 35.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-41.7 - 35.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-18.5 + 32.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-14.0 - 38.6i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (0.615 + 3.49i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-27.5 - 32.7i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 47.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-61.1 - 10.7i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (8.50 - 7.13i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (105. - 38.2i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (90.9 + 52.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (48.5 + 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (104. + 37.9i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-44.8 + 123. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (87.8 - 50.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (28.7 + 163. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.68120896595461735863013475233, −10.24587941240116643853102500043, −9.708445794066436892828945434824, −8.812417319362107954288588910796, −7.62400112373125303550828141489, −6.29453511413847550962914643969, −5.85811484436876806711119170203, −4.28802895507419532791393150597, −2.95023051210767912274097613002, −1.44400958588006683515416582636,
1.13172510547586994605501351689, 2.75427113057416093095229942583, 4.20026426368025955972143036447, 5.41416157780235514044375775681, 6.51091596001600912031951055573, 7.29040959475896911939174293028, 8.749105615654370947379034407090, 9.554567206884539680606947071147, 10.15507932966860545426869197387, 11.43693556949495924502602405881
