| L(s) = 1 | + 2·4-s + 3·5-s − 14·11-s − 5·16-s − 16·19-s + 6·20-s + 5·25-s − 4·29-s − 2·31-s − 4·41-s − 28·44-s − 42·55-s − 20·59-s + 24·61-s − 20·64-s − 4·71-s − 32·76-s − 34·79-s − 15·80-s + 20·89-s − 48·95-s + 10·100-s + 36·101-s + 30·109-s − 8·116-s + 95·121-s − 4·124-s + ⋯ |
| L(s) = 1 | + 4-s + 1.34·5-s − 4.22·11-s − 5/4·16-s − 3.67·19-s + 1.34·20-s + 25-s − 0.742·29-s − 0.359·31-s − 0.624·41-s − 4.22·44-s − 5.66·55-s − 2.60·59-s + 3.07·61-s − 5/2·64-s − 0.474·71-s − 3.67·76-s − 3.82·79-s − 1.67·80-s + 2.11·89-s − 4.92·95-s + 100-s + 3.58·101-s + 2.87·109-s − 0.742·116-s + 8.63·121-s − 0.359·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366311287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366311287\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 33 T^{2} + 536 T^{4} - 33 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 846 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 950 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 21 T^{2} + 1952 T^{4} - 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 2904 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 193 T^{2} + 14856 T^{4} - 193 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 10 T + 110 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 17006 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T + 110 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 17 T + 222 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 5814 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 170 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.86927819880008607321345877625, −6.51092187021437559607256727027, −6.43410180247871852000312606233, −6.26098378890219883023573557553, −6.10048431651004580242421919663, −5.76411745561887244511703311168, −5.63567115529696747756206491348, −5.27452693295993777412933709381, −5.25606559183777277517886688200, −4.92019331034461521528071545333, −4.64308328006121757131520599320, −4.43958314593660061371970823524, −4.43631103149416162867149898285, −3.89042212788785104759087022990, −3.65493315569520899609632918764, −3.13008792036218194031245019249, −2.79818578297069133206855960314, −2.77137083396353360471567422121, −2.65395071942177938444270968474, −2.05895958927615268588247061142, −1.98867669896816222221939369017, −1.91202766230064766523254954016, −1.77986386512124333407816739865, −0.50637135791050198570641533203, −0.33356172391887945225620828877,
0.33356172391887945225620828877, 0.50637135791050198570641533203, 1.77986386512124333407816739865, 1.91202766230064766523254954016, 1.98867669896816222221939369017, 2.05895958927615268588247061142, 2.65395071942177938444270968474, 2.77137083396353360471567422121, 2.79818578297069133206855960314, 3.13008792036218194031245019249, 3.65493315569520899609632918764, 3.89042212788785104759087022990, 4.43631103149416162867149898285, 4.43958314593660061371970823524, 4.64308328006121757131520599320, 4.92019331034461521528071545333, 5.25606559183777277517886688200, 5.27452693295993777412933709381, 5.63567115529696747756206491348, 5.76411745561887244511703311168, 6.10048431651004580242421919663, 6.26098378890219883023573557553, 6.43410180247871852000312606233, 6.51092187021437559607256727027, 6.86927819880008607321345877625
