| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 0.918i·5-s − 10.4i·7-s + 2.82·8-s + 1.29i·10-s − 1.55i·11-s + 9.98·13-s − 14.8i·14-s + 4.00·16-s + 6.91i·17-s − 33.2i·19-s + 1.83i·20-s − 2.20i·22-s + (−13.6 − 18.4i)23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.183i·5-s − 1.49i·7-s + 0.353·8-s + 0.129i·10-s − 0.141i·11-s + 0.768·13-s − 1.05i·14-s + 0.250·16-s + 0.406i·17-s − 1.75i·19-s + 0.0918i·20-s − 0.100i·22-s + (−0.594 − 0.804i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.34894 - 1.18460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34894 - 1.18460i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (13.6 + 18.4i)T \) |
| good | 5 | \( 1 - 0.918iT - 25T^{2} \) |
| 7 | \( 1 + 10.4iT - 49T^{2} \) |
| 11 | \( 1 + 1.55iT - 121T^{2} \) |
| 13 | \( 1 - 9.98T + 169T^{2} \) |
| 17 | \( 1 - 6.91iT - 289T^{2} \) |
| 19 | \( 1 + 33.2iT - 361T^{2} \) |
| 29 | \( 1 - 7.68T + 841T^{2} \) |
| 31 | \( 1 + 3.45T + 961T^{2} \) |
| 37 | \( 1 - 24.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 46.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 60.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 45.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 73.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 40.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 69.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 33.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 94.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 82.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 32.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 25.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 114. iT - 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.82394232445910858461189346373, −10.39453420804975810194557108340, −9.038810392826655216593430798701, −7.889507473805582755279620529145, −6.93011266477445162759658904235, −6.24881254794590769881379986921, −4.77682654448227212549805689938, −4.00802345045925616823941882517, −2.82169245448700160981654220164, −0.974917487559090904866888459665,
1.75674905800591662987561793444, 3.03228089431441195018055311318, 4.24608944712933215814925298572, 5.63344951804906908503315709178, 5.92448050910635205331760350038, 7.35917491823392366225059958784, 8.442693534130488592241313885978, 9.225968161437489237310315620559, 10.36235393316502017191714752746, 11.38309905146069723116414562852
