| L(s) = 1 | − 2·2-s − 3-s + 3·4-s − 5-s + 2·6-s − 2·7-s − 4·8-s − 4·9-s + 2·10-s + 11-s − 3·12-s − 4·13-s + 4·14-s + 15-s + 5·16-s − 2·17-s + 8·18-s − 3·20-s + 2·21-s − 2·22-s + 4·24-s + 2·25-s + 8·26-s + 6·27-s − 6·28-s − 9·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.41·8-s − 4/3·9-s + 0.632·10-s + 0.301·11-s − 0.866·12-s − 1.10·13-s + 1.06·14-s + 0.258·15-s + 5/4·16-s − 0.485·17-s + 1.88·18-s − 0.670·20-s + 0.436·21-s − 0.426·22-s + 0.816·24-s + 2/5·25-s + 1.56·26-s + 1.15·27-s − 1.13·28-s − 1.67·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 11 T + 101 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 91 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 19 T + 183 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 75 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 154 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 157 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 170 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 213 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−7.81784849833013615421812029452, −7.71544335753327203037239361564, −7.49538614827024797094531922078, −7.06631064335179957221502224417, −6.78922616052701716393139350602, −6.28291658204746859060331700978, −5.87671376642184026479684332246, −5.76028908054702902151704525055, −5.24964151870805833671423690576, −4.98471665984429149272767935565, −4.06441892255196783054115126643, −3.99304065592862589958949204865, −3.42521407221471281275732985167, −2.86979970585991765331797119764, −2.41334529935236143644434361499, −2.30685413658062913296336691243, −1.45364016571174925183457017542, −0.77134490922761403662279850040, 0, 0,
0.77134490922761403662279850040, 1.45364016571174925183457017542, 2.30685413658062913296336691243, 2.41334529935236143644434361499, 2.86979970585991765331797119764, 3.42521407221471281275732985167, 3.99304065592862589958949204865, 4.06441892255196783054115126643, 4.98471665984429149272767935565, 5.24964151870805833671423690576, 5.76028908054702902151704525055, 5.87671376642184026479684332246, 6.28291658204746859060331700978, 6.78922616052701716393139350602, 7.06631064335179957221502224417, 7.49538614827024797094531922078, 7.71544335753327203037239361564, 7.81784849833013615421812029452
