| L(s) = 1 | + 3.18i·2-s − 0.724i·3-s − 6.17·4-s + 7.66i·5-s + 2.31·6-s − 10.1i·7-s − 6.93i·8-s + 8.47·9-s − 24.4·10-s − 2.43·11-s + 4.47i·12-s + 18.1·13-s + 32.3·14-s + 5.55·15-s − 2.56·16-s − 1.13·17-s + ⋯ |
| L(s) = 1 | + 1.59i·2-s − 0.241i·3-s − 1.54·4-s + 1.53i·5-s + 0.385·6-s − 1.44i·7-s − 0.867i·8-s + 0.941·9-s − 2.44·10-s − 0.221·11-s + 0.372i·12-s + 1.39·13-s + 2.30·14-s + 0.370·15-s − 0.160·16-s − 0.0668·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.508398 + 0.942177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508398 + 0.942177i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-23.6 - 35.9i)T \) |
| good | 2 | \( 1 - 3.18iT - 4T^{2} \) |
| 3 | \( 1 + 0.724iT - 9T^{2} \) |
| 5 | \( 1 - 7.66iT - 25T^{2} \) |
| 7 | \( 1 + 10.1iT - 49T^{2} \) |
| 11 | \( 1 + 2.43T + 121T^{2} \) |
| 13 | \( 1 - 18.1T + 169T^{2} \) |
| 17 | \( 1 + 1.13T + 289T^{2} \) |
| 19 | \( 1 + 12.3iT - 361T^{2} \) |
| 23 | \( 1 + 24.2T + 529T^{2} \) |
| 29 | \( 1 + 35.5iT - 841T^{2} \) |
| 31 | \( 1 + 12.9T + 961T^{2} \) |
| 37 | \( 1 - 41.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 65.0T + 1.68e3T^{2} \) |
| 47 | \( 1 - 51.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 56.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 88.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 65.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 45.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 63.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 8.80iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 31.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 68.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 5.90iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 100.T + 9.40e3T^{2} \) |
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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
−15.90837262134642793302888387730, −15.18901798823119060397294870095, −13.85678581426571478671542068310, −13.51832605971891400059162528119, −11.08759172015533384725513832655, −10.03517640499595481846533351122, −7.973921695708962753195757935974, −7.05165582860087801247629754626, −6.32326823478957323977116468781, −4.05029658287028016652525092089,
1.65841435123056432683290311835, 3.96291013845544232158186303894, 5.45903340111289014273680447667, 8.588204109117985501052580824096, 9.266206650096693770906066391707, 10.59167370490043841868817313334, 12.08262585553958769693851959855, 12.53479604982410125782912073732, 13.53657218177087252897099965035, 15.58477659947896799062421316842
