| L(s) = 1 | + (−2.23 − 2i)3-s + 2i·7-s + (1.00 + 8.94i)9-s − 13.4i·11-s + 8i·13-s + 13.4·17-s − 34·19-s + (4 − 4.47i)21-s + 40.2·23-s + (15.6 − 22.0i)27-s + 40.2i·29-s − 14·31-s + (−26.8 + 30.0i)33-s − 56i·37-s + (16 − 17.8i)39-s + ⋯ |
| L(s) = 1 | + (−0.745 − 0.666i)3-s + 0.285i·7-s + (0.111 + 0.993i)9-s − 1.21i·11-s + 0.615i·13-s + 0.789·17-s − 1.78·19-s + (0.190 − 0.212i)21-s + 1.74·23-s + (0.579 − 0.814i)27-s + 1.38i·29-s − 0.451·31-s + (−0.813 + 0.909i)33-s − 1.51i·37-s + (0.410 − 0.458i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.082155460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082155460\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.23 + 2i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 49T^{2} \) |
| 11 | \( 1 + 13.4iT - 121T^{2} \) |
| 13 | \( 1 - 8iT - 169T^{2} \) |
| 17 | \( 1 - 13.4T + 289T^{2} \) |
| 19 | \( 1 + 34T + 361T^{2} \) |
| 23 | \( 1 - 40.2T + 529T^{2} \) |
| 29 | \( 1 - 40.2iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 + 56iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 40.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 13.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 46T + 3.72e3T^{2} \) |
| 67 | \( 1 - 32iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 22T + 6.24e3T^{2} \) |
| 83 | \( 1 + 120.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 122iT - 9.40e3T^{2} \) |
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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
−8.966871207797252302520352072425, −8.695037663519649271486130641321, −7.46814581794463854526652822177, −6.82999365294134592449129142063, −5.89911202009456025601811472563, −5.33536529099693282095377787580, −4.16656189511005046018473942612, −2.89535148860365289166359237299, −1.67402543037542901523687387110, −0.42670323305732760533765521618,
1.04131617643235049155565538138, 2.63878534003948261062056868727, 3.92890121185152428876901230082, 4.63860788961005539345229045868, 5.46641954491667331468390487436, 6.44596269020314768667014101283, 7.16835619064361030536459042822, 8.191028394378118645097944041845, 9.178612705150110240335058346964, 9.983512641457918606262529684426
