L(s) = 1 | + i·5-s − 1.26·7-s + 3.46i·11-s − 3.46i·13-s − 3.46·17-s − 2i·19-s + 8.19·23-s − 25-s + 9.46·31-s − 1.26i·35-s + 6i·37-s − 2.53·41-s + 10.1i·43-s − 8.19·47-s − 5.39·49-s + ⋯ |
L(s) = 1 | + 0.447i·5-s − 0.479·7-s + 1.04i·11-s − 0.960i·13-s − 0.840·17-s − 0.458i·19-s + 1.70·23-s − 0.200·25-s + 1.69·31-s − 0.214i·35-s + 0.986i·37-s − 0.396·41-s + 1.55i·43-s − 1.19·47-s − 0.770·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.196397080\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.196397080\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 4.73iT - 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−9.017730713935180070592652277678, −8.228419053375635418483009860739, −7.37294549808494223126421982299, −6.73830414974396240417962854000, −6.13353844082001422107185555589, −4.95009263289322932831623936628, −4.45805377874823238036654968758, −3.09328418056973568516646821151, −2.68061071083011225059821945717, −1.21742491526225121391326462421,
0.40610297949479450680656394433, 1.70508584190718889663602764046, 2.89838561759075302840972019915, 3.74248122743724503558144945626, 4.67567359779411195895236525783, 5.41762659699721921527286959763, 6.46967290842681220755621965876, 6.78479144257994616752568653028, 7.947349591902545305202353906579, 8.675633811973169694877733674988