| L(s) = 1 | + (−1.99 − 1.72i)2-s + (−2.55 + 1.35i)3-s + (0.738 + 4.90i)4-s + (−1.87 + 1.21i)5-s + (7.43 + 1.69i)6-s + (0.323 + 2.62i)7-s + (4.14 − 6.60i)8-s + (3.02 − 4.43i)9-s + (5.84 + 0.800i)10-s + (−1.90 + 1.29i)11-s + (−8.51 − 11.5i)12-s + (−4.66 + 1.63i)13-s + (3.87 − 5.80i)14-s + (3.15 − 5.64i)15-s + (−10.1 + 3.14i)16-s + (0.669 − 1.53i)17-s + ⋯ |
| L(s) = 1 | + (−1.41 − 1.21i)2-s + (−1.47 + 0.780i)3-s + (0.369 + 2.45i)4-s + (−0.839 + 0.543i)5-s + (3.03 + 0.693i)6-s + (0.122 + 0.992i)7-s + (1.46 − 2.33i)8-s + (1.00 − 1.47i)9-s + (1.84 + 0.253i)10-s + (−0.573 + 0.390i)11-s + (−2.45 − 3.33i)12-s + (−1.29 + 0.453i)13-s + (1.03 − 1.55i)14-s + (0.815 − 1.45i)15-s + (−2.54 + 0.785i)16-s + (0.162 − 0.372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00220711 - 0.00775922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00220711 - 0.00775922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.87 - 1.21i)T \) |
| 7 | \( 1 + (-0.323 - 2.62i)T \) |
| good | 2 | \( 1 + (1.99 + 1.72i)T + (0.298 + 1.97i)T^{2} \) |
| 3 | \( 1 + (2.55 - 1.35i)T + (1.68 - 2.47i)T^{2} \) |
| 11 | \( 1 + (1.90 - 1.29i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (4.66 - 1.63i)T + (10.1 - 8.10i)T^{2} \) |
| 17 | \( 1 + (-0.669 + 1.53i)T + (-11.5 - 12.4i)T^{2} \) |
| 19 | \( 1 + (-1.69 - 2.93i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.46 + 0.639i)T + (15.6 - 16.8i)T^{2} \) |
| 29 | \( 1 + (3.85 + 3.07i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (3.18 + 1.83i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.10 + 1.55i)T + (10.9 - 35.3i)T^{2} \) |
| 41 | \( 1 + (3.25 - 0.743i)T + (36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (-4.67 + 2.93i)T + (18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-4.62 + 5.37i)T + (-7.00 - 46.4i)T^{2} \) |
| 53 | \( 1 + (0.614 + 0.453i)T + (15.6 + 50.6i)T^{2} \) |
| 59 | \( 1 + (-4.75 - 4.40i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-1.54 + 10.2i)T + (-58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-2.65 - 9.90i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (4.39 + 5.50i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-5.08 - 5.91i)T + (-10.8 + 72.1i)T^{2} \) |
| 79 | \( 1 + (-7.65 + 4.41i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.25 - 6.43i)T + (-64.8 - 51.7i)T^{2} \) |
| 89 | \( 1 + (5.65 + 3.85i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + (11.3 + 11.3i)T + 97iT^{2} \) |
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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
−11.55693253310665712637443010583, −10.78453180566442875493450942246, −9.968034987303434183030776354623, −9.312740701642655008468904302988, −7.937728825165838625986250023719, −7.00001695945321777058810302506, −5.31797529981449917301543533689, −3.99609377631480874181159212638, −2.47894803484952850995563902627, −0.01506115680482500756845595000,
1.05746067692891536331120459168, 4.87879981123706225835997229126, 5.60374203193689198611558366188, 6.95637290385780360132989114670, 7.40690419335896565861392251855, 8.123497360750006875155792551169, 9.514332128106212816861761824884, 10.67689243411798724526753867054, 11.11622175564386680927516985383, 12.32123103828402930202841974616
