| L(s) = 1 | − 2-s + 1.04·3-s + 4-s + 1.64·5-s − 1.04·6-s − 2.76·7-s − 8-s − 1.90·9-s − 1.64·10-s + 4.88·11-s + 1.04·12-s + 2.76·14-s + 1.71·15-s + 16-s + 4.84·17-s + 1.90·18-s − 19-s + 1.64·20-s − 2.88·21-s − 4.88·22-s − 5.33·23-s − 1.04·24-s − 2.29·25-s − 5.12·27-s − 2.76·28-s + 1.03·29-s − 1.71·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.603·3-s + 0.5·4-s + 0.734·5-s − 0.426·6-s − 1.04·7-s − 0.353·8-s − 0.636·9-s − 0.519·10-s + 1.47·11-s + 0.301·12-s + 0.738·14-s + 0.443·15-s + 0.250·16-s + 1.17·17-s + 0.449·18-s − 0.229·19-s + 0.367·20-s − 0.629·21-s − 1.04·22-s − 1.11·23-s − 0.213·24-s − 0.459·25-s − 0.986·27-s − 0.522·28-s + 0.192·29-s − 0.313·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800076085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800076085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 17 | \( 1 - 4.84T + 17T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 + 0.813T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 + 6.71T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 - 0.592T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 + 4.85T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.081283982114816780980066778186, −7.47766379015955207139372917680, −6.52617056199270294593410857908, −6.08217970061076058022214338677, −5.54898306249795570181729971330, −4.06428062627857262980031417415, −3.48755811244871898666786817797, −2.62782804746877165795073630111, −1.86485249442428366946363900108, −0.74742339023143757087240570668,
0.74742339023143757087240570668, 1.86485249442428366946363900108, 2.62782804746877165795073630111, 3.48755811244871898666786817797, 4.06428062627857262980031417415, 5.54898306249795570181729971330, 6.08217970061076058022214338677, 6.52617056199270294593410857908, 7.47766379015955207139372917680, 8.081283982114816780980066778186
