| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 4·7-s − 4·8-s + 3·9-s − 4·10-s + 6·12-s − 6·13-s + 8·14-s + 4·15-s + 5·16-s − 4·17-s − 6·18-s + 2·19-s + 6·20-s − 8·21-s − 6·23-s − 8·24-s + 3·25-s + 12·26-s + 4·27-s − 12·28-s + 6·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.51·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.73·12-s − 1.66·13-s + 2.13·14-s + 1.03·15-s + 5/4·16-s − 0.970·17-s − 1.41·18-s + 0.458·19-s + 1.34·20-s − 1.74·21-s − 1.25·23-s − 1.63·24-s + 3/5·25-s + 2.35·26-s + 0.769·27-s − 2.26·28-s + 1.11·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 92 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 104 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 120 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 59 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 34 T + 480 T^{2} + 34 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.286433230858081471655928255959, −8.250466072332683760518381754917, −7.57514212429952952011378828367, −7.49373956898189556419139347128, −6.77009175372342197415293575043, −6.75832216327628477449279477655, −6.23769644447077194338752075487, −6.20893173714196632091830305283, −5.27909776161272245534405475970, −5.11196585130936830155056680307, −4.31884984647027034770105781437, −4.15896884527577305908593563706, −3.17139296967985312060598144135, −3.05975479113967359778543676153, −2.65303259731027719132882956057, −2.36931756005792574965742831479, −1.58247959897517413610129991301, −1.42525740998277458022868181350, 0, 0,
1.42525740998277458022868181350, 1.58247959897517413610129991301, 2.36931756005792574965742831479, 2.65303259731027719132882956057, 3.05975479113967359778543676153, 3.17139296967985312060598144135, 4.15896884527577305908593563706, 4.31884984647027034770105781437, 5.11196585130936830155056680307, 5.27909776161272245534405475970, 6.20893173714196632091830305283, 6.23769644447077194338752075487, 6.75832216327628477449279477655, 6.77009175372342197415293575043, 7.49373956898189556419139347128, 7.57514212429952952011378828367, 8.250466072332683760518381754917, 8.286433230858081471655928255959
