| L(s) = 1 | + (−1.60 − 2.14i)2-s + (−1.49 + 1.49i)3-s + (−1.45 + 4.96i)4-s + (1.95 − 1.08i)5-s + (5.61 + 0.807i)6-s + (0.692 − 3.18i)7-s + (7.97 − 2.97i)8-s − 1.48i·9-s + (−5.47 − 2.43i)10-s + (−3.23 + 0.717i)11-s + (−5.25 − 9.62i)12-s + (2.28 + 4.19i)13-s + (−7.94 + 3.62i)14-s + (−1.29 + 4.55i)15-s + (−10.4 − 6.72i)16-s + (−0.142 − 1.99i)17-s + ⋯ |
| L(s) = 1 | + (−1.13 − 1.51i)2-s + (−0.864 + 0.864i)3-s + (−0.729 + 2.48i)4-s + (0.873 − 0.487i)5-s + (2.29 + 0.329i)6-s + (0.261 − 1.20i)7-s + (2.81 − 1.05i)8-s − 0.495i·9-s + (−1.73 − 0.771i)10-s + (−0.976 + 0.216i)11-s + (−1.51 − 2.77i)12-s + (0.634 + 1.16i)13-s + (−2.12 + 0.969i)14-s + (−0.333 + 1.17i)15-s + (−2.61 − 1.68i)16-s + (−0.0346 − 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580287 - 0.309891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580287 - 0.309891i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.95 + 1.08i)T \) |
| 11 | \( 1 + (3.23 - 0.717i)T \) |
| good | 2 | \( 1 + (1.60 + 2.14i)T + (-0.563 + 1.91i)T^{2} \) |
| 3 | \( 1 + (1.49 - 1.49i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.692 + 3.18i)T + (-6.36 - 2.90i)T^{2} \) |
| 13 | \( 1 + (-2.28 - 4.19i)T + (-7.02 + 10.9i)T^{2} \) |
| 17 | \( 1 + (0.142 + 1.99i)T + (-16.8 + 2.41i)T^{2} \) |
| 19 | \( 1 + (-4.26 - 4.92i)T + (-2.70 + 18.8i)T^{2} \) |
| 23 | \( 1 + (1.78 - 0.388i)T + (20.9 - 9.55i)T^{2} \) |
| 29 | \( 1 + (2.17 + 2.51i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (6.48 - 1.90i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-10.2 - 5.61i)T + (20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-9.95 - 1.43i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.11 + 2.65i)T + (32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (-3.82 + 5.11i)T + (-13.2 - 45.0i)T^{2} \) |
| 53 | \( 1 + (-0.146 + 0.674i)T + (-48.2 - 22.0i)T^{2} \) |
| 59 | \( 1 + (-6.88 + 0.989i)T + (56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (0.642 - 0.0923i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.96 + 4.46i)T + (18.8 - 64.2i)T^{2} \) |
| 71 | \( 1 + (-5.10 - 5.88i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.944 - 4.34i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (2.10 + 4.60i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-3.90 - 0.850i)T + (75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-2.71 - 2.35i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (5.66 - 2.11i)T + (73.3 - 63.5i)T^{2} \) |
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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
−10.53507494551767289089748595172, −9.746282049553726749588045069152, −9.450808559636369810837554488483, −8.177354772056300218970163135947, −7.33704681730423179579370177268, −5.70195401483515000323937625512, −4.52816983954613616163706520456, −3.84181220048537120267122882113, −2.18915527924189589149387542702, −0.936704202190084911136976032722,
0.861498476800185669744114793268, 2.39264991460392188713669469708, 5.35099321016358505292371710208, 5.70947742573648198509926220224, 6.16932672066369840540708490534, 7.32529815359519303470864628672, 7.84880862045049907838645390186, 8.982055242607920657631440645989, 9.548118694462732389771080089632, 10.81722229507269999956981733091
