L(s) = 1 | + (−0.5 − 0.866i)5-s + (−0.366 + 0.633i)7-s + (−0.866 + 1.5i)11-s + (0.732 + 1.26i)13-s + 1.26·17-s + 2.46·19-s + (−1.73 − 3i)23-s + (−0.499 + 0.866i)25-s + (−2.13 + 3.69i)29-s + (3.96 + 6.86i)31-s + 0.732·35-s + 4.19·37-s + (−0.401 − 0.696i)41-s + (−3.36 + 5.83i)43-s + (−2.36 + 4.09i)47-s + ⋯ |
L(s) = 1 | + (−0.223 − 0.387i)5-s + (−0.138 + 0.239i)7-s + (−0.261 + 0.452i)11-s + (0.203 + 0.351i)13-s + 0.307·17-s + 0.565·19-s + (−0.361 − 0.625i)23-s + (−0.0999 + 0.173i)25-s + (−0.396 + 0.686i)29-s + (0.711 + 1.23i)31-s + 0.123·35-s + 0.689·37-s + (−0.0627 − 0.108i)41-s + (−0.513 + 0.889i)43-s + (−0.345 + 0.597i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.432342949\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.432342949\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.366 - 0.633i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.866 - 1.5i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.732 - 1.26i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 + (1.73 + 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.13 - 3.69i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.96 - 6.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + (0.401 + 0.696i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.36 - 5.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.36 - 4.09i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + (2.13 + 3.69i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.19 - 12.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.803T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + (3.19 - 5.53i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.56 + 7.90i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + (-1.36 + 2.36i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.493133802806327159463564168495, −8.690352662901520253185779700631, −7.990005110703853969109635760557, −7.12792129416723213662016262824, −6.30176041197663038177562247278, −5.31374656234015113196068737547, −4.58282860240332098426835435899, −3.56189849732701379700125085281, −2.48259924825547099527862024690, −1.16461105691845869204672437455,
0.63753473210227509541304620065, 2.24514148451576566058258975297, 3.34233844967448321672221284080, 4.05821550950106060479881964453, 5.30459267325605611810041189566, 5.99339741029452014291650471570, 6.93690953840875704167988151208, 7.76686114058653824518998291710, 8.304757765886104323230902006686, 9.414996520761329599683378121763