L(s) = 1 | − 2·3-s − 4-s − 2·5-s + 9-s − 4·11-s + 2·12-s + 4·15-s + 4·16-s − 4·17-s + 4·19-s + 2·20-s − 8·23-s + 25-s + 2·27-s − 8·29-s + 12·31-s + 8·33-s − 36-s − 4·37-s + 8·41-s + 4·44-s − 2·45-s + 8·47-s − 8·48-s + 8·51-s + 16·53-s + 8·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.577·12-s + 1.03·15-s + 16-s − 0.970·17-s + 0.917·19-s + 0.447·20-s − 1.66·23-s + 1/5·25-s + 0.384·27-s − 1.48·29-s + 2.15·31-s + 1.39·33-s − 1/6·36-s − 0.657·37-s + 1.24·41-s + 0.603·44-s − 0.298·45-s + 1.16·47-s − 1.15·48-s + 1.12·51-s + 2.19·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8368947513\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8368947513\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_4\times C_2$ | \( 1 + 4 T + 10 T^{2} - 64 T^{3} - 261 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T - 6 T^{2} + 64 T^{3} - 181 T^{4} + 64 p T^{5} - 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 66 T^{2} - 192 T^{3} + 659 T^{4} - 192 p T^{5} + 66 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 304 T^{3} - 1957 T^{4} - 304 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 34 T^{2} + 512 T^{3} - 4317 T^{4} + 512 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 2 p T^{2} - 704 T^{3} + 6123 T^{4} - 704 p T^{5} + 2 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 38 T^{2} - 2037 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 16 T + 66 T^{2} + 704 T^{3} + 12083 T^{4} + 704 p T^{5} + 66 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 30 T^{2} - 512 T^{3} - 2461 T^{4} - 512 p T^{5} - 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.42289921571481749444944829550, −7.27973830093201096079364969245, −6.95101937834725531772088365094, −6.79837944367599767568796311919, −6.45786922261183585903407800110, −6.40602359417542271624968473252, −5.86717321981884763299159969498, −5.72006197049598269715505202336, −5.51138104049294860479350456507, −5.44619458310482459477891982327, −5.26624244602216208744115235114, −4.74399789842339470393692055365, −4.59550781865808286512922844218, −4.32091423705100474460901939964, −4.02903891975598314108292533636, −3.91910520157997141901908222291, −3.57972843822771351418203090734, −3.13267073328124617067415515575, −2.98638916249248001400267725906, −2.47796861404182658085371440212, −2.20616257945777101579512402355, −1.94521928647690216833674890916, −1.21152634180676958180354746852, −0.63089790121514005596669228616, −0.47124607906426607212310816895,
0.47124607906426607212310816895, 0.63089790121514005596669228616, 1.21152634180676958180354746852, 1.94521928647690216833674890916, 2.20616257945777101579512402355, 2.47796861404182658085371440212, 2.98638916249248001400267725906, 3.13267073328124617067415515575, 3.57972843822771351418203090734, 3.91910520157997141901908222291, 4.02903891975598314108292533636, 4.32091423705100474460901939964, 4.59550781865808286512922844218, 4.74399789842339470393692055365, 5.26624244602216208744115235114, 5.44619458310482459477891982327, 5.51138104049294860479350456507, 5.72006197049598269715505202336, 5.86717321981884763299159969498, 6.40602359417542271624968473252, 6.45786922261183585903407800110, 6.79837944367599767568796311919, 6.95101937834725531772088365094, 7.27973830093201096079364969245, 7.42289921571481749444944829550