| L(s) = 1 | − 2·3-s + 6·4-s + 4·5-s + 2·9-s − 10·11-s − 12·12-s − 8·15-s + 19·16-s − 16·17-s + 24·20-s − 12·23-s + 8·25-s + 4·27-s + 8·29-s + 8·31-s + 20·33-s + 12·36-s + 8·41-s − 60·44-s + 8·45-s − 4·47-s − 38·48-s + 32·51-s − 40·55-s − 48·60-s + 20·61-s + 36·64-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3·4-s + 1.78·5-s + 2/3·9-s − 3.01·11-s − 3.46·12-s − 2.06·15-s + 19/4·16-s − 3.88·17-s + 5.36·20-s − 2.50·23-s + 8/5·25-s + 0.769·27-s + 1.48·29-s + 1.43·31-s + 3.48·33-s + 2·36-s + 1.24·41-s − 9.04·44-s + 1.19·45-s − 0.583·47-s − 5.48·48-s + 4.48·51-s − 5.39·55-s − 6.19·60-s + 2.56·61-s + 9/2·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.665323601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665323601\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 180 T^{3} + 599 T^{4} + 180 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1354 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 120 T^{3} + 254 T^{4} - 120 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 136 T^{3} + 382 T^{4} - 136 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 2542 T^{4} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 216 T^{3} + 1262 T^{4} - 216 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 4438 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 6843 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7510 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 1780 T^{3} + 15058 T^{4} - 1780 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 124 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 420 T^{3} + 9799 T^{4} + 420 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 60 T^{3} - 601 T^{4} - 60 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 26998 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 28 T + 363 T^{2} + 28 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 396 T^{3} - 18814 T^{4} + 396 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.99367109978534769776681225635, −6.98614607763300866586562908954, −6.97787678640604418719072564106, −6.35653020490243297214071747829, −6.34112442202005338175790663282, −6.28434354537388803991964887358, −6.25521292052051616837337738225, −5.70711578848950911798152264842, −5.68372386862732304180403548106, −5.27294408492664243061335220504, −5.24863154785204799578442155059, −4.75099409975859832717689863662, −4.59926697628781139572114520059, −4.28981115022411115080835959688, −4.19667901205818217356661625859, −3.51212859876705616562244671190, −2.92659910549256691678781855990, −2.91187319007356576490069486577, −2.57342262619309884353217740176, −2.26683661036255902545138237714, −2.16879364792152622433460794199, −2.16487272795190921765323480040, −1.71150544167818824438674631727, −1.07724244026199176817814348752, −0.36859198158612203429157943122,
0.36859198158612203429157943122, 1.07724244026199176817814348752, 1.71150544167818824438674631727, 2.16487272795190921765323480040, 2.16879364792152622433460794199, 2.26683661036255902545138237714, 2.57342262619309884353217740176, 2.91187319007356576490069486577, 2.92659910549256691678781855990, 3.51212859876705616562244671190, 4.19667901205818217356661625859, 4.28981115022411115080835959688, 4.59926697628781139572114520059, 4.75099409975859832717689863662, 5.24863154785204799578442155059, 5.27294408492664243061335220504, 5.68372386862732304180403548106, 5.70711578848950911798152264842, 6.25521292052051616837337738225, 6.28434354537388803991964887358, 6.34112442202005338175790663282, 6.35653020490243297214071747829, 6.97787678640604418719072564106, 6.98614607763300866586562908954, 6.99367109978534769776681225635
