| L(s) = 1 | + 2·4-s + 8·11-s + 3·16-s − 32·29-s − 8·41-s + 16·44-s − 2·49-s + 16·59-s + 24·61-s + 12·64-s + 8·71-s − 24·89-s + 40·101-s − 40·109-s − 64·116-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 16·164-s + 167-s + 28·169-s + ⋯ |
| L(s) = 1 | + 4-s + 2.41·11-s + 3/4·16-s − 5.94·29-s − 1.24·41-s + 2.41·44-s − 2/7·49-s + 2.08·59-s + 3.07·61-s + 3/2·64-s + 0.949·71-s − 2.54·89-s + 3.98·101-s − 3.83·109-s − 5.94·116-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s − 1.24·164-s + 0.0773·167-s + 2.15·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7490275269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7490275269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 + 4 T^{2} + 70 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4726 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 886 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 116 T^{2} + 10822 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_4$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 40166 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.85329403574493223044627047580, −6.46427732159324793843380970929, −6.31360344724133285924808681745, −6.27973680996660126595340740270, −5.81607308120844197657008752173, −5.50007349180539755116699059936, −5.41517212914690456979566204726, −5.39180608208496519223926105476, −5.26962678418123954728303725645, −4.58633996598587717213865152763, −4.57439908541044768565528067486, −3.98968959882373509877399624182, −3.87221734926877695632809105934, −3.85375577704468597653801541684, −3.57633060670762066066954313065, −3.38448795743330742643513596726, −3.33188895437703631700515446817, −2.44846236965363946029852847867, −2.36683198387829876158923926619, −2.17290118948740712111499433321, −2.05219627483637834166423560279, −1.41633441648611835895558654391, −1.21098970600976099045330974551, −1.20701618278576920330468950728, −0.13883502307462636961419284135,
0.13883502307462636961419284135, 1.20701618278576920330468950728, 1.21098970600976099045330974551, 1.41633441648611835895558654391, 2.05219627483637834166423560279, 2.17290118948740712111499433321, 2.36683198387829876158923926619, 2.44846236965363946029852847867, 3.33188895437703631700515446817, 3.38448795743330742643513596726, 3.57633060670762066066954313065, 3.85375577704468597653801541684, 3.87221734926877695632809105934, 3.98968959882373509877399624182, 4.57439908541044768565528067486, 4.58633996598587717213865152763, 5.26962678418123954728303725645, 5.39180608208496519223926105476, 5.41517212914690456979566204726, 5.50007349180539755116699059936, 5.81607308120844197657008752173, 6.27973680996660126595340740270, 6.31360344724133285924808681745, 6.46427732159324793843380970929, 6.85329403574493223044627047580
