| L(s) = 1 | + (0.825 − 0.367i)2-s + (−0.453 + 1.67i)3-s + (−0.792 + 0.879i)4-s + (−1.78 − 1.35i)5-s + (0.239 + 1.54i)6-s + (−1.53 + 2.65i)7-s + (−0.888 + 2.73i)8-s + (−2.58 − 1.51i)9-s + (−1.96 − 0.461i)10-s + (−1.20 + 0.534i)11-s + (−1.11 − 1.72i)12-s + (4.56 + 2.03i)13-s + (−0.289 + 2.75i)14-s + (3.06 − 2.36i)15-s + (0.0240 + 0.229i)16-s + (1.36 − 4.18i)17-s + ⋯ |
| L(s) = 1 | + (0.583 − 0.259i)2-s + (−0.261 + 0.965i)3-s + (−0.396 + 0.439i)4-s + (−0.796 − 0.604i)5-s + (0.0978 + 0.631i)6-s + (−0.579 + 1.00i)7-s + (−0.314 + 0.967i)8-s + (−0.862 − 0.505i)9-s + (−0.621 − 0.145i)10-s + (−0.361 + 0.161i)11-s + (−0.320 − 0.497i)12-s + (1.26 + 0.563i)13-s + (−0.0773 + 0.736i)14-s + (0.792 − 0.610i)15-s + (0.00602 + 0.0572i)16-s + (0.329 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.416734 + 0.784082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.416734 + 0.784082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.453 - 1.67i)T \) |
| 5 | \( 1 + (1.78 + 1.35i)T \) |
| good | 2 | \( 1 + (-0.825 + 0.367i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (1.53 - 2.65i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.20 - 0.534i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-4.56 - 2.03i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.36 + 4.18i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.05 - 3.24i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.595 - 5.66i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-6.43 - 1.36i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (5.33 - 1.13i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-5.90 + 4.29i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.45 - 0.647i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (2.24 - 3.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.49 - 1.38i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (2.18 + 6.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.22 + 1.43i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (11.2 - 4.98i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (5.39 - 1.14i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (2.00 + 6.15i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.08 - 3.69i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (2.76 + 0.586i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-8.07 - 8.96i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-5.99 - 4.35i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-16.6 - 3.53i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.34005319199562804788125863995, −11.82008851321134251189877744488, −10.96897015839256085446553960322, −9.397916125458532665774480201212, −8.909033422255779173259826796163, −7.87927418867474041528973643558, −5.95745584678339964622949186032, −5.05229031085686535107303470502, −3.98990857748493558164377764428, −3.09642042315819788165892093405,
0.65454547480687447998511670266, 3.23714012402577403223393507818, 4.38893846071269409856261543822, 6.02276385053114428635311127323, 6.58462206432205928466294933123, 7.67995110659289427349971357633, 8.676369810206488226011518640885, 10.49242585062811262056583475091, 10.76170613143239419308803255434, 12.20829639728182151089631914536
