| L(s) = 1 | + 1.73i·3-s + 2·5-s + 6.92i·7-s − 2.99·9-s + 6.92i·11-s − 2·13-s + 3.46i·15-s + 10·17-s + 20.7i·19-s − 11.9·21-s + 27.7i·23-s − 21·25-s − 5.19i·27-s + 26·29-s − 6.92i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.400·5-s + 0.989i·7-s − 0.333·9-s + 0.629i·11-s − 0.153·13-s + 0.230i·15-s + 0.588·17-s + 1.09i·19-s − 0.571·21-s + 1.20i·23-s − 0.839·25-s − 0.192i·27-s + 0.896·29-s − 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.04165 + 1.04165i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04165 + 1.04165i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| good | 5 | \( 1 - 2T + 25T^{2} \) |
| 7 | \( 1 - 6.92iT - 49T^{2} \) |
| 11 | \( 1 - 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 2T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 - 26T + 841T^{2} \) |
| 31 | \( 1 + 6.92iT - 961T^{2} \) |
| 37 | \( 1 + 26T + 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74T + 2.80e3T^{2} \) |
| 59 | \( 1 + 90.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−12.30994236711842367607352152945, −11.75708638393646185810657213220, −10.34026127358801225949876956283, −9.677196034768099341392142727470, −8.717165311987638300053998517589, −7.53248001597013587796042673047, −5.98611054186061833859331948800, −5.20911113389778063738607086778, −3.68636738717169391107319437857, −2.10540305371607791461657824528,
0.899907469215992686316824256879, 2.76962669435860102907115344980, 4.39567593106682269030443981156, 5.86724718930888547116028605221, 6.90152327176605236378331248794, 7.87472050326695064117571857597, 9.003237493038277317007575898471, 10.21374067539048479110659762489, 11.03401133810739354441756124658, 12.14645749425567440806275740897
