| L(s) = 1 | + 4·3-s + 6·9-s − 8·11-s − 16·13-s − 8·23-s − 4·27-s − 32·33-s − 16·37-s − 64·39-s − 24·47-s + 4·49-s − 40·59-s − 8·61-s − 32·69-s + 8·73-s − 37·81-s + 24·83-s + 24·97-s − 48·99-s − 24·107-s + 8·109-s − 64·111-s − 96·117-s + 12·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s − 2.41·11-s − 4.43·13-s − 1.66·23-s − 0.769·27-s − 5.57·33-s − 2.63·37-s − 10.2·39-s − 3.50·47-s + 4/7·49-s − 5.20·59-s − 1.02·61-s − 3.85·69-s + 0.936·73-s − 4.11·81-s + 2.63·83-s + 2.43·97-s − 4.82·99-s − 2.32·107-s + 0.766·109-s − 6.07·111-s − 8.87·117-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1328602202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1328602202\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2470 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7030 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
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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
−6.27252396754547514216866403856, −6.20852425735929537285651097059, −6.17047068807792562231000348071, −5.50084691938631028927136927193, −5.40720923817624870582214367697, −5.22213057623578311361987813295, −5.05561126672492639463850772118, −4.85277752892982403815487200932, −4.79106298650751840079549812919, −4.57289417327083522790150986416, −4.23960758897556861023301515707, −3.95535886112904238088271592051, −3.59117101114215285322094148512, −3.33544435665379058196536461592, −3.26829942207667292478373804827, −2.97472140384437327300531710623, −2.83820329128576610840026874727, −2.54201503937574905516244074026, −2.30091224363711569770726135859, −2.22148096966224090232157613673, −2.00084160188907229968147827147, −1.57926587556318444857645786072, −1.55114436472951823245019376638, −0.22526136689745842713208368664, −0.13030152238924639891241159026,
0.13030152238924639891241159026, 0.22526136689745842713208368664, 1.55114436472951823245019376638, 1.57926587556318444857645786072, 2.00084160188907229968147827147, 2.22148096966224090232157613673, 2.30091224363711569770726135859, 2.54201503937574905516244074026, 2.83820329128576610840026874727, 2.97472140384437327300531710623, 3.26829942207667292478373804827, 3.33544435665379058196536461592, 3.59117101114215285322094148512, 3.95535886112904238088271592051, 4.23960758897556861023301515707, 4.57289417327083522790150986416, 4.79106298650751840079549812919, 4.85277752892982403815487200932, 5.05561126672492639463850772118, 5.22213057623578311361987813295, 5.40720923817624870582214367697, 5.50084691938631028927136927193, 6.17047068807792562231000348071, 6.20852425735929537285651097059, 6.27252396754547514216866403856
