L(s) = 1 | + 2-s − 1.88·3-s + 4-s − 0.545·5-s − 1.88·6-s − 1.80·7-s + 8-s + 0.567·9-s − 0.545·10-s − 2.25·11-s − 1.88·12-s − 2.31·13-s − 1.80·14-s + 1.02·15-s + 16-s + 2.80·17-s + 0.567·18-s − 8.14·19-s − 0.545·20-s + 3.41·21-s − 2.25·22-s + 6.33·23-s − 1.88·24-s − 4.70·25-s − 2.31·26-s + 4.59·27-s − 1.80·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.09·3-s + 0.5·4-s − 0.243·5-s − 0.771·6-s − 0.682·7-s + 0.353·8-s + 0.189·9-s − 0.172·10-s − 0.680·11-s − 0.545·12-s − 0.641·13-s − 0.482·14-s + 0.265·15-s + 0.250·16-s + 0.681·17-s + 0.133·18-s − 1.86·19-s − 0.121·20-s + 0.744·21-s − 0.480·22-s + 1.31·23-s − 0.385·24-s − 0.940·25-s − 0.453·26-s + 0.884·27-s − 0.341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.193692763\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.193692763\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + 1.88T + 3T^{2} \) |
| 5 | \( 1 + 0.545T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 2.25T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.642T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 7.87T + 67T^{2} \) |
| 71 | \( 1 + 0.432T + 71T^{2} \) |
| 73 | \( 1 - 0.290T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 4.29T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
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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
−8.665531309087116128996139740898, −7.999340571846794292820416366335, −6.82608103963551244355280065686, −6.56369564519256846494994211516, −5.62734195101689751054396279580, −5.02686174581020144925757359101, −4.27763910226565608427470942111, −3.18036353326527913271956598915, −2.33621797558626678967400902314, −0.61645020285375665264133519468,
0.61645020285375665264133519468, 2.33621797558626678967400902314, 3.18036353326527913271956598915, 4.27763910226565608427470942111, 5.02686174581020144925757359101, 5.62734195101689751054396279580, 6.56369564519256846494994211516, 6.82608103963551244355280065686, 7.999340571846794292820416366335, 8.665531309087116128996139740898