| L(s) = 1 | − 3.12·3-s − 9.83·5-s + 0.781·9-s − 20.9·11-s + 7.40·13-s + 30.7·15-s − 32.3·19-s + 71.7·25-s + 25.7·27-s + 29·29-s − 36.8·31-s + 65.4·33-s − 23.1·39-s + 85.9·43-s − 7.68·45-s − 29.6·47-s + 49·49-s − 105.·53-s + 206.·55-s + 101.·57-s − 72.8·65-s − 224.·75-s − 136.·79-s − 87.4·81-s − 90.6·87-s + 115.·93-s + 317.·95-s + ⋯ |
| L(s) = 1 | − 1.04·3-s − 1.96·5-s + 0.0867·9-s − 1.90·11-s + 0.569·13-s + 2.05·15-s − 1.70·19-s + 2.87·25-s + 0.952·27-s + 29-s − 1.18·31-s + 1.98·33-s − 0.594·39-s + 1.99·43-s − 0.170·45-s − 0.630·47-s + 0.999·49-s − 1.99·53-s + 3.74·55-s + 1.77·57-s − 1.12·65-s − 2.99·75-s − 1.72·79-s − 1.07·81-s − 1.04·87-s + 1.23·93-s + 3.34·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2840156485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2840156485\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 3.12T + 9T^{2} \) |
| 5 | \( 1 + 9.83T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 20.9T + 121T^{2} \) |
| 13 | \( 1 - 7.40T + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 32.3T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 31 | \( 1 + 36.8T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 85.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 29.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 105.T + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 136.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.81969944312434293777585854001, −10.59788945244848129942218268544, −8.700374856967576405429772900200, −8.061362762343505503010063868250, −7.24433783773485912552202745089, −6.11381131626714466397563842712, −4.99643582806553520496978751300, −4.17935813104296612376022523189, −2.87444309833045021536103406128, −0.39005126510023331440663463006,
0.39005126510023331440663463006, 2.87444309833045021536103406128, 4.17935813104296612376022523189, 4.99643582806553520496978751300, 6.11381131626714466397563842712, 7.24433783773485912552202745089, 8.061362762343505503010063868250, 8.700374856967576405429772900200, 10.59788945244848129942218268544, 10.81969944312434293777585854001
