| L(s) = 1 | + (−1.26 − 0.460i)2-s + (0.5 + 2.83i)3-s + (−0.141 − 0.118i)4-s + (0.673 − 0.565i)5-s + (0.673 − 3.82i)6-s + (−0.173 + 0.300i)7-s + (1.47 + 2.54i)8-s + (−4.97 + 1.80i)9-s + (−1.11 + 0.405i)10-s + (1.11 + 1.92i)11-s + (0.266 − 0.460i)12-s + (−0.446 + 2.53i)13-s + (0.358 − 0.300i)14-s + (1.93 + 1.62i)15-s + (−0.624 − 3.54i)16-s + (−0.439 − 0.160i)17-s + ⋯ |
| L(s) = 1 | + (−0.895 − 0.325i)2-s + (0.288 + 1.63i)3-s + (−0.0707 − 0.0593i)4-s + (0.301 − 0.252i)5-s + (0.275 − 1.55i)6-s + (−0.0656 + 0.113i)7-s + (0.520 + 0.901i)8-s + (−1.65 + 0.603i)9-s + (−0.352 + 0.128i)10-s + (0.335 + 0.581i)11-s + (0.0768 − 0.133i)12-s + (−0.123 + 0.703i)13-s + (0.0957 − 0.0803i)14-s + (0.500 + 0.420i)15-s + (−0.156 − 0.885i)16-s + (−0.106 − 0.0388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.335091 + 0.634936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335091 + 0.634936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (1.26 + 0.460i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 2.83i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.673 + 0.565i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.300i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.446 - 2.53i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (0.439 + 0.160i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.06 + 1.73i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (6.46 - 2.35i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.55 - 6.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 + (0.429 + 2.43i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.98 + 2.50i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.85 + 2.49i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.17 - 1.82i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (5.92 + 2.15i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.99 - 5.86i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-7.21 + 2.62i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-7.12 + 5.98i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.241 - 1.36i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.05 - 11.6i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.41 + 12.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.78 + 10.1i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-8.88 - 3.23i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.22565208723575958746274161939, −10.55338022970024239502931438308, −9.787165051944286603041647970722, −9.104075846751142012853462767862, −8.766187558467209415969032676391, −7.31094077819523205632252559891, −5.56938748190301954940139173176, −4.75457814350976061340249065684, −3.71607669660541443665334882054, −2.00338936878068971259901676162,
0.64137899474596304733884782368, 2.15710785971080710162627869216, 3.69837937368677750960996013181, 5.77814727141087674862971643151, 6.65774052992137412588924595465, 7.55403807932232263751118415785, 8.099220063033932941087467426750, 8.996046118076194863094650617980, 9.951456919291481931215239827786, 11.11780978882793402264136326539
