L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (2.5 + 0.866i)7-s + (1 − 1.73i)9-s + (1 + 1.73i)11-s + 0.999·15-s + (−2 − 3.46i)17-s + (−1 + 1.73i)19-s + (0.500 + 2.59i)21-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + 5·27-s + 9·29-s + (2 + 3.46i)31-s + (−0.999 + 1.73i)33-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.944 + 0.327i)7-s + (0.333 − 0.577i)9-s + (0.301 + 0.522i)11-s + 0.258·15-s + (−0.485 − 0.840i)17-s + (−0.229 + 0.397i)19-s + (0.109 + 0.566i)21-s + (0.104 − 0.180i)23-s + (−0.0999 − 0.173i)25-s + 0.962·27-s + 1.67·29-s + (0.359 + 0.622i)31-s + (−0.174 + 0.301i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.87184 + 0.238535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.87184 + 0.238535i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 7.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (6 + 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 18T + 97T^{2} \) |
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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
−10.61400094245281204296903874310, −9.901372826978615428649650507682, −8.964172176595218572298651761736, −8.438941620644930682805941783654, −7.22284430521483368562770377650, −6.23693928829709708671890587212, −4.88178119080158725589911936399, −4.40007195155935534396074380940, −2.91669772162565995869674692252, −1.44369515060516476287777848657,
1.43813149596822599097363937277, 2.57579544874835584876761979961, 4.07099546771574391429523996449, 5.08524211226001843257308566396, 6.35473580551789476797455451055, 7.15284113747364488551167311343, 8.145345776914433445990037187422, 8.656787668983808565073152161985, 10.03491452436992801544487347217, 10.75918225059477247521260318799