L(s) = 1 | − 2-s + 2.23·3-s + 4-s − 5-s − 2.23·6-s + 7-s − 8-s + 2.00·9-s + 10-s + 2.23·12-s + 13-s − 14-s − 2.23·15-s + 16-s + 3.23·17-s − 2.00·18-s + 2.23·19-s − 20-s + 2.23·21-s − 8.23·23-s − 2.23·24-s + 25-s − 26-s − 2.23·27-s + 28-s − 4.47·29-s + 2.23·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.447·5-s − 0.912·6-s + 0.377·7-s − 0.353·8-s + 0.666·9-s + 0.316·10-s + 0.645·12-s + 0.277·13-s − 0.267·14-s − 0.577·15-s + 0.250·16-s + 0.784·17-s − 0.471·18-s + 0.512·19-s − 0.223·20-s + 0.487·21-s − 1.71·23-s − 0.456·24-s + 0.200·25-s − 0.196·26-s − 0.430·27-s + 0.188·28-s − 0.830·29-s + 0.408·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 0.763T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 2.76T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 - 5.76T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.59678279233050086686461187014, −7.24700723622687158554385952442, −6.15271164089432478869951377365, −5.47059377605500960068677965096, −4.41164144425328976219731921180, −3.54391758655861140139800648002, −3.15105347525637893361597702738, −2.06557513668394933611581132643, −1.47945003216174396944052996806, 0,
1.47945003216174396944052996806, 2.06557513668394933611581132643, 3.15105347525637893361597702738, 3.54391758655861140139800648002, 4.41164144425328976219731921180, 5.47059377605500960068677965096, 6.15271164089432478869951377365, 7.24700723622687158554385952442, 7.59678279233050086686461187014