| L(s) = 1 | − 1.41·2-s − 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (−4.63 + 5.24i)7-s − 2.82·8-s − 2.99·9-s + 11.7·11-s − 3.46i·12-s + 24.8i·13-s + (6.54 − 7.42i)14-s + 4.00·16-s − 7.26i·17-s + 4.24·18-s − 23.0i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (−0.661 + 0.749i)7-s − 0.353·8-s − 0.333·9-s + 1.06·11-s − 0.288i·12-s + 1.90i·13-s + (0.467 − 0.530i)14-s + 0.250·16-s − 0.427i·17-s + 0.235·18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3873444984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3873444984\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (4.63 - 5.24i)T \) |
| good | 11 | \( 1 - 11.7T + 121T^{2} \) |
| 13 | \( 1 - 24.8iT - 169T^{2} \) |
| 17 | \( 1 + 7.26iT - 289T^{2} \) |
| 19 | \( 1 + 23.0iT - 361T^{2} \) |
| 23 | \( 1 - 26.4T + 529T^{2} \) |
| 29 | \( 1 + 57.0T + 841T^{2} \) |
| 31 | \( 1 + 10.2iT - 961T^{2} \) |
| 37 | \( 1 + 14.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 51.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 64.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 22.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 91.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 71.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 45.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 17.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 77.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 6.15iT - 9.40e3T^{2} \) |
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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
−9.539107536606343181385834024449, −9.218956193679476878562808674291, −8.718036339993119083904083458878, −7.34904022552865963875186895451, −6.78435475857773219669242968563, −6.20286556317234935278051907667, −4.94228508251570302359038030706, −3.60284082810970779662024144014, −2.41397038743785662293694150560, −1.43983607357490877823393290149,
0.15544459701964950860229213540, 1.48306049619028936099992929209, 3.28236471646141287076235047007, 3.67736410624606623930044867301, 5.17214976845721588344215762981, 6.08000319345669788873885402150, 6.97155938892780299078787289698, 7.87891418259307510076452800864, 8.623908744071091073319617482176, 9.621253178956509834745026121164
