| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 2·5-s + 4·6-s + 2·8-s + 9-s + 4·10-s − 4·11-s − 2·12-s + 4·15-s − 4·16-s − 2·18-s − 2·20-s + 8·22-s − 4·23-s − 4·24-s + 25-s + 2·27-s + 16·29-s − 8·30-s + 4·31-s + 2·32-s + 8·33-s + 36-s − 4·40-s − 24·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s + 0.707·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.577·12-s + 1.03·15-s − 16-s − 0.471·18-s − 0.447·20-s + 1.70·22-s − 0.834·23-s − 0.816·24-s + 1/5·25-s + 0.384·27-s + 2.97·29-s − 1.46·30-s + 0.718·31-s + 0.353·32-s + 1.39·33-s + 1/6·36-s − 0.632·40-s − 3.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5304954486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5304954486\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} + 8 T^{3} + 279 T^{4} + 8 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 184 T^{3} - 1201 T^{4} + 184 p T^{5} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 72 T^{2} + 3815 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 120 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 32 T^{2} + 184 T^{3} - 657 T^{4} + 184 p T^{5} - 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 496 T^{3} - 3333 T^{4} - 496 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T + 62 T^{2} + 432 T^{3} - 5253 T^{4} + 432 p T^{5} + 62 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 18 T^{2} - 48 T^{3} + 3371 T^{4} - 48 p T^{5} + 18 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T - 24 T^{2} + 376 T^{3} - 3961 T^{4} + 376 p T^{5} - 24 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 102 T^{2} - 112 T^{3} + 7427 T^{4} - 112 p T^{5} - 102 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 30 T^{2} - 5341 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 94 T^{2} - 272 T^{3} + 2755 T^{4} - 272 p T^{5} - 94 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.01496126336090822623716916298, −6.55644702096843847254803467350, −6.31224961022658230080042850929, −6.28423519298494638676699618511, −5.83863340725434712325054509560, −5.82036260820054687766156706558, −5.50061450971152352497789218676, −5.22769316478609199825667658685, −5.00993183306606390268070222634, −4.81510848255930719762375182890, −4.57712114761661048296297274895, −4.41639242852346952976546110091, −4.27284539593529618821790422922, −3.75591419435204632760761275178, −3.59804297133878023242134720295, −3.32787646149041249381882683378, −3.10427964429370067315110536756, −2.59711669259207851235398988256, −2.44675397965694623883318618618, −2.06023741250319486092324017762, −1.99958523418173970202159386959, −1.11112538662217581010700687359, −1.03625650107283008920231458933, −0.54717203242586045290212328605, −0.41727357623473776281040838839,
0.41727357623473776281040838839, 0.54717203242586045290212328605, 1.03625650107283008920231458933, 1.11112538662217581010700687359, 1.99958523418173970202159386959, 2.06023741250319486092324017762, 2.44675397965694623883318618618, 2.59711669259207851235398988256, 3.10427964429370067315110536756, 3.32787646149041249381882683378, 3.59804297133878023242134720295, 3.75591419435204632760761275178, 4.27284539593529618821790422922, 4.41639242852346952976546110091, 4.57712114761661048296297274895, 4.81510848255930719762375182890, 5.00993183306606390268070222634, 5.22769316478609199825667658685, 5.50061450971152352497789218676, 5.82036260820054687766156706558, 5.83863340725434712325054509560, 6.28423519298494638676699618511, 6.31224961022658230080042850929, 6.55644702096843847254803467350, 7.01496126336090822623716916298
