| L(s) = 1 | + 2.23·2-s + 1.00·4-s + 12i·7-s − 6.70·8-s − 15.6i·11-s + 11i·13-s + 26.8i·14-s − 19·16-s − 24.5·17-s − 16·19-s − 35i·22-s − 6.70·23-s + 24.5i·26-s + 12.0i·28-s + 11.1i·29-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.250·4-s + 1.71i·7-s − 0.838·8-s − 1.42i·11-s + 0.846i·13-s + 1.91i·14-s − 1.18·16-s − 1.44·17-s − 0.842·19-s − 1.59i·22-s − 0.291·23-s + 0.946i·26-s + 0.428i·28-s + 0.385i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.114012011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114012011\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 4T^{2} \) |
| 7 | \( 1 - 12iT - 49T^{2} \) |
| 11 | \( 1 + 15.6iT - 121T^{2} \) |
| 13 | \( 1 - 11iT - 169T^{2} \) |
| 17 | \( 1 + 24.5T + 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 6.70T + 529T^{2} \) |
| 29 | \( 1 - 11.1iT - 841T^{2} \) |
| 31 | \( 1 - 9T + 961T^{2} \) |
| 37 | \( 1 - 2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 22.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 91.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 84.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 35.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 92iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 69T + 6.24e3T^{2} \) |
| 83 | \( 1 - 67.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 58iT - 9.40e3T^{2} \) |
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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
−11.21088056793959822595237714049, −9.613136192244273786530660816741, −8.697348971422801102915604361135, −8.498847518348697487576183387080, −6.53555193565653340634172632841, −6.11869904897658907517994842589, −5.19452577060841136216835691590, −4.29541371217482944127954573293, −3.10750298830243457930090893558, −2.19829143708921708949860734784,
0.25396794020931642065627429725, 2.17194787352217376368142389779, 3.64865488292381478238155582125, 4.35652478559814472496570391935, 4.98138754132855397742070104577, 6.37862357233321412500270760248, 7.01892386299669057085477747484, 7.983834865627317804678040674127, 9.155647766423463310096323834061, 10.18528771782617306485964948077
