| L(s) = 1 | + 0.821i·2-s + 3.32·4-s + 2.23i·5-s + 0.837·7-s + 6.01i·8-s − 1.83·10-s − 14.3i·11-s − 21.8·13-s + 0.688i·14-s + 8.35·16-s − 23.5i·17-s − 6.32·19-s + 7.43i·20-s + 11.8·22-s + 38.8i·23-s + ⋯ |
| L(s) = 1 | + 0.410i·2-s + 0.831·4-s + 0.447i·5-s + 0.119·7-s + 0.752i·8-s − 0.183·10-s − 1.30i·11-s − 1.67·13-s + 0.0491i·14-s + 0.521·16-s − 1.38i·17-s − 0.332·19-s + 0.371i·20-s + 0.536·22-s + 1.69i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.18398 + 0.376314i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18398 + 0.376314i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 2 | \( 1 - 0.821iT - 4T^{2} \) |
| 7 | \( 1 - 0.837T + 49T^{2} \) |
| 11 | \( 1 + 14.3iT - 121T^{2} \) |
| 13 | \( 1 + 21.8T + 169T^{2} \) |
| 17 | \( 1 + 23.5iT - 289T^{2} \) |
| 19 | \( 1 + 6.32T + 361T^{2} \) |
| 23 | \( 1 - 38.8iT - 529T^{2} \) |
| 29 | \( 1 - 0.266iT - 841T^{2} \) |
| 31 | \( 1 - 30.2T + 961T^{2} \) |
| 37 | \( 1 - 9.53T + 1.36e3T^{2} \) |
| 41 | \( 1 - 19.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 22.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 48.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 77.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 68.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 114.T + 9.40e3T^{2} \) |
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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
−15.73147363151331804569480838857, −14.70109824690313244814912240321, −13.71220517853582774153299799100, −11.92853916267964030932021728046, −11.15832404584836242295139046215, −9.709099867688240486140627612717, −7.923046945115465127414880215665, −6.86503121228493291441844040760, −5.39483661754537902531414157674, −2.81733722071413148947982709004,
2.21734075795209065088395487819, 4.56909271600230675211091365389, 6.56829261372947477653770287766, 7.895392803542448984445601914335, 9.712953329781814439164688409725, 10.67208408599646578480143953317, 12.32422120165479537685595355840, 12.54613221378673121671461144928, 14.61234066231656025087461010139, 15.35858299883370848340100199427
