| L(s) = 1 | − 16·11-s − 8·13-s − 8·23-s − 2·25-s − 8·37-s + 16·49-s − 32·59-s + 40·61-s − 8·73-s − 48·83-s − 24·97-s − 48·107-s − 40·109-s + 120·121-s + 127-s + 131-s + 137-s + 139-s + 128·143-s + 149-s + 151-s + 157-s + 163-s + 167-s − 8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 4.82·11-s − 2.21·13-s − 1.66·23-s − 2/5·25-s − 1.31·37-s + 16/7·49-s − 4.16·59-s + 5.12·61-s − 0.936·73-s − 5.26·83-s − 2.43·97-s − 4.64·107-s − 3.83·109-s + 10.9·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.7·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08293638460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08293638460\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 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 | $C_4\times C_2$ | \( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 418 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7030 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11830 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 16 T + 180 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4614 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.79794682135626334160360276781, −6.77213987707723257656505294708, −6.48144415644044816824336908772, −5.90921358044068909375144391480, −5.72482668957035523704622096188, −5.61786075098206753695476386655, −5.47246576712210034803522664290, −5.25294005742020447763546785498, −5.20555527959876291940503360636, −4.85873886610452850591929924530, −4.77872358348729169092522851647, −4.26330982678663996564535477361, −4.09645573458607914689342383210, −3.94321465814082114241693158567, −3.75469928657225169008237401970, −2.94443855828417120921757167260, −2.87781869918281055013556588308, −2.70807452517557273754609515357, −2.68623303248484705423205622993, −2.36924244560887649771519773585, −2.13403615606214718347597235670, −1.59552567414014799363337751330, −1.40958643605283253068621518841, −0.33777597376407490990255275440, −0.12641982256569041312251386921,
0.12641982256569041312251386921, 0.33777597376407490990255275440, 1.40958643605283253068621518841, 1.59552567414014799363337751330, 2.13403615606214718347597235670, 2.36924244560887649771519773585, 2.68623303248484705423205622993, 2.70807452517557273754609515357, 2.87781869918281055013556588308, 2.94443855828417120921757167260, 3.75469928657225169008237401970, 3.94321465814082114241693158567, 4.09645573458607914689342383210, 4.26330982678663996564535477361, 4.77872358348729169092522851647, 4.85873886610452850591929924530, 5.20555527959876291940503360636, 5.25294005742020447763546785498, 5.47246576712210034803522664290, 5.61786075098206753695476386655, 5.72482668957035523704622096188, 5.90921358044068909375144391480, 6.48144415644044816824336908772, 6.77213987707723257656505294708, 6.79794682135626334160360276781
