L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−1.32 − 2.29i)7-s + (−0.499 − 0.866i)9-s + (0.822 − 1.42i)11-s − 2.64·13-s + 0.999·15-s + (0.822 − 1.42i)17-s + (−4.14 − 7.18i)19-s + 2.64·21-s + (−0.822 − 1.42i)23-s + (−0.499 + 0.866i)25-s + 0.999·27-s + 7.64·29-s + (2.14 − 3.71i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.499 − 0.866i)7-s + (−0.166 − 0.288i)9-s + (0.248 − 0.429i)11-s − 0.733·13-s + 0.258·15-s + (0.199 − 0.345i)17-s + (−0.951 − 1.64i)19-s + 0.577·21-s + (−0.171 − 0.297i)23-s + (−0.0999 + 0.173i)25-s + 0.192·27-s + 1.41·29-s + (0.385 − 0.667i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.555043 - 0.591382i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.555043 - 0.591382i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.32 + 2.29i)T \) |
good | 11 | \( 1 + (-0.822 + 1.42i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + (-0.822 + 1.42i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.14 + 7.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.822 + 1.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 + (-2.14 + 3.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.322 + 0.559i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.93T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.64 - 2.85i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.46 - 9.47i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.322 - 0.559i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.14 + 1.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + (-7.11 - 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.91345419405648271009729866043, −10.08726987741506826181576256842, −9.269224367286378008446015164527, −8.312329249471234876190076533129, −7.11615356215034793730304401592, −6.30017843459664662762383890961, −4.90103082890823506593209326670, −4.19528950172189039769452603207, −2.85068171649318596338223853751, −0.52885149077074766655734548929,
1.93556318074068897418215183173, 3.23440761245805510457951021020, 4.69085356341645749707963176647, 5.98212531102319681356605782646, 6.59823363616323770365279907284, 7.74755754013065176432037022718, 8.561384353522931026361829436276, 9.782435076819715593067800619018, 10.44933674963216971402297660901, 11.67252884373997739638875708613