| L(s) = 1 | − 2·3-s − 2·4-s − 8·7-s + 9-s + 6·11-s + 4·12-s + 3·16-s + 16·21-s + 5·25-s − 4·27-s + 16·28-s − 12·33-s − 2·36-s + 16·37-s + 30·41-s − 12·44-s − 12·47-s − 6·48-s + 12·49-s + 12·53-s − 8·63-s − 4·64-s − 2·67-s + 10·73-s − 10·75-s − 48·77-s − 4·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s − 3.02·7-s + 1/3·9-s + 1.80·11-s + 1.15·12-s + 3/4·16-s + 3.49·21-s + 25-s − 0.769·27-s + 3.02·28-s − 2.08·33-s − 1/3·36-s + 2.63·37-s + 4.68·41-s − 1.80·44-s − 1.75·47-s − 0.866·48-s + 12/7·49-s + 1.64·53-s − 1.00·63-s − 1/2·64-s − 0.244·67-s + 1.17·73-s − 1.15·75-s − 5.47·77-s − 4/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29986576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2885874323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2885874323\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - p T^{2} + 9 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 633 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 162 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 101 T^{2} + 4185 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 1653 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 5169 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 5 T + 105 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 11973 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 20478 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−10.91633864637183167844453857725, −10.60372255719571875243541001959, −10.04927674140541890680663199485, −9.736437178033479842865496534244, −9.700846868135914281300537808868, −9.566375257743002105551904355025, −9.079728672022026253730412723331, −8.899194145903140863029741060291, −8.822974507471760822583719598981, −7.84117264190993255120497014889, −7.78092525267977142108962169786, −7.48824916527483325333338914894, −6.70025664105353145292165270198, −6.59542030368139108942676692192, −6.33636746403213413824536341217, −6.02007695550803456404350041039, −5.97832402100446085088432750213, −5.32315482422903812575679653540, −4.84351231959469134717944299118, −4.34737798094927207572296928875, −3.85325996899596242988396231979, −3.77761671350164625409554474866, −3.04893253332420279706449546079, −2.62509516017638999425379875924, −0.860193621872294310796872716453,
0.860193621872294310796872716453, 2.62509516017638999425379875924, 3.04893253332420279706449546079, 3.77761671350164625409554474866, 3.85325996899596242988396231979, 4.34737798094927207572296928875, 4.84351231959469134717944299118, 5.32315482422903812575679653540, 5.97832402100446085088432750213, 6.02007695550803456404350041039, 6.33636746403213413824536341217, 6.59542030368139108942676692192, 6.70025664105353145292165270198, 7.48824916527483325333338914894, 7.78092525267977142108962169786, 7.84117264190993255120497014889, 8.822974507471760822583719598981, 8.899194145903140863029741060291, 9.079728672022026253730412723331, 9.566375257743002105551904355025, 9.700846868135914281300537808868, 9.736437178033479842865496534244, 10.04927674140541890680663199485, 10.60372255719571875243541001959, 10.91633864637183167844453857725
