| L(s) = 1 | + (−0.193 − 1.22i)2-s + (−2.26 + 1.15i)3-s + (0.443 − 0.144i)4-s + (−0.536 + 2.17i)5-s + (1.85 + 2.54i)6-s + (−1.57 + 3.09i)7-s + (−1.38 − 2.72i)8-s + (2.04 − 2.80i)9-s + (2.75 + 0.235i)10-s + (−0.839 + 0.839i)12-s + (−0.730 + 0.115i)13-s + (4.08 + 1.32i)14-s + (−1.29 − 5.54i)15-s + (−2.30 + 1.67i)16-s + (0.609 + 0.0965i)17-s + (−3.83 − 1.95i)18-s + ⋯ |
| L(s) = 1 | + (−0.136 − 0.864i)2-s + (−1.30 + 0.666i)3-s + (0.221 − 0.0720i)4-s + (−0.239 + 0.970i)5-s + (0.755 + 1.04i)6-s + (−0.595 + 1.16i)7-s + (−0.490 − 0.962i)8-s + (0.680 − 0.936i)9-s + (0.872 + 0.0744i)10-s + (−0.242 + 0.242i)12-s + (−0.202 + 0.0320i)13-s + (1.09 + 0.354i)14-s + (−0.333 − 1.43i)15-s + (−0.576 + 0.418i)16-s + (0.147 + 0.0234i)17-s + (−0.903 − 0.460i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00477539 - 0.0559073i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00477539 - 0.0559073i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.536 - 2.17i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.193 + 1.22i)T + (-1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (2.26 - 1.15i)T + (1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.57 - 3.09i)T + (-4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (0.730 - 0.115i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.609 - 0.0965i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.971 + 2.99i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (4.30 + 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + (0.896 + 2.75i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.45 + 1.78i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 1.04i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (0.970 + 0.315i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.07 - 4.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.492 + 0.967i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (0.671 + 4.24i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (7.03 - 2.28i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.20 - 3.03i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (9.39 - 9.39i)T - 67iT^{2} \) |
| 71 | \( 1 + (2.92 - 2.12i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.479 - 0.244i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (9.87 + 7.17i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.50 + 15.8i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (-14.1 + 2.24i)T + (92.2 - 29.9i)T^{2} \) |
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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
−10.20197131478393134474026918794, −10.00898565475159207807648110618, −8.898237566314249740711711585943, −7.32417391727791983070708831748, −6.21312862596538890722562223224, −5.94943078246410663321591037320, −4.52062167277447245392403314583, −3.26396428073926631415214454852, −2.31285954560625098339733832694, −0.03756557204717741776926716647,
1.46905403254193843034872927734, 3.68160925745217650806267781262, 5.04191707861407062937990378660, 5.79924248275664288290991426750, 6.60143219520385011559966598965, 7.42723657657188132861385862436, 7.935517057986753916913539477034, 9.223882138676371084537951467042, 10.31158666332846598393687075625, 11.20661771365838151967028490463
