| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s − 7-s − 8-s + 6·9-s + 10-s − 3·12-s + 13-s + 14-s + 3·15-s + 16-s + 3·17-s − 6·18-s − 6·19-s − 20-s + 3·21-s − 4·23-s + 3·24-s − 4·25-s − 26-s − 9·27-s − 28-s − 2·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.866·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s − 1.41·18-s − 1.37·19-s − 0.223·20-s + 0.654·21-s − 0.834·23-s + 0.612·24-s − 4/5·25-s − 0.196·26-s − 1.73·27-s − 0.188·28-s − 0.371·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3146 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3146 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.174465989121661359437511477893, −7.52260855659566213355764232106, −6.69700065044511212992831284617, −6.06963680351175114271015390023, −5.58007316868712318029783390276, −4.43660190737283567298107688401, −3.76677853005733776527358303686, −2.25144343695204805750190070200, −0.973879971324260460585952773727, 0,
0.973879971324260460585952773727, 2.25144343695204805750190070200, 3.76677853005733776527358303686, 4.43660190737283567298107688401, 5.58007316868712318029783390276, 6.06963680351175114271015390023, 6.69700065044511212992831284617, 7.52260855659566213355764232106, 8.174465989121661359437511477893
