L(s) = 1 | + 2-s + 1.31·3-s + 4-s + 5-s + 1.31·6-s − 4.24·7-s + 8-s − 1.26·9-s + 10-s − 1.63·11-s + 1.31·12-s − 13-s − 4.24·14-s + 1.31·15-s + 16-s + 5.84·17-s − 1.26·18-s − 1.36·19-s + 20-s − 5.59·21-s − 1.63·22-s − 8.85·23-s + 1.31·24-s + 25-s − 26-s − 5.61·27-s − 4.24·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.760·3-s + 0.5·4-s + 0.447·5-s + 0.537·6-s − 1.60·7-s + 0.353·8-s − 0.421·9-s + 0.316·10-s − 0.493·11-s + 0.380·12-s − 0.277·13-s − 1.13·14-s + 0.340·15-s + 0.250·16-s + 1.41·17-s − 0.298·18-s − 0.313·19-s + 0.223·20-s − 1.22·21-s − 0.348·22-s − 1.84·23-s + 0.268·24-s + 0.200·25-s − 0.196·26-s − 1.08·27-s − 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 17 | \( 1 - 5.84T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 1.07T + 29T^{2} \) |
| 37 | \( 1 + 0.732T + 37T^{2} \) |
| 41 | \( 1 + 0.539T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 - 0.441T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 5.44T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6.52T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.956502971095051446052536296753, −7.39093028407292327251934905318, −6.32558534389199236075355140765, −5.95535182354785545674893321655, −5.20119131429327527232259355469, −3.99120210024961262411455580506, −3.28597854696934119982049556220, −2.78649906347016164662345601203, −1.84900083085015256002188108955, 0,
1.84900083085015256002188108955, 2.78649906347016164662345601203, 3.28597854696934119982049556220, 3.99120210024961262411455580506, 5.20119131429327527232259355469, 5.95535182354785545674893321655, 6.32558534389199236075355140765, 7.39093028407292327251934905318, 7.956502971095051446052536296753