L(s) = 1 | − 2·4-s − 8·5-s − 8·11-s + 3·16-s + 16·20-s + 38·25-s + 16·29-s + 4·31-s − 8·41-s + 16·44-s + 4·49-s + 64·55-s + 24·59-s − 8·61-s − 4·64-s − 24·80-s + 8·89-s − 76·100-s + 24·109-s − 32·116-s + 12·121-s − 8·124-s − 136·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | − 4-s − 3.57·5-s − 2.41·11-s + 3/4·16-s + 3.57·20-s + 38/5·25-s + 2.97·29-s + 0.718·31-s − 1.24·41-s + 2.41·44-s + 4/7·49-s + 8.62·55-s + 3.12·59-s − 1.02·61-s − 1/2·64-s − 2.68·80-s + 0.847·89-s − 7.59·100-s + 2.29·109-s − 2.97·116-s + 1.09·121-s − 0.718·124-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6187256129\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6187256129\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
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 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1590 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 4694 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4966 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25654 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.28490646401135145631564581838, −5.98340164432540046319805181445, −5.91200830580157344034137548353, −5.32336525289576685220608852465, −5.25325373515817224139226357954, −5.05201035346064229935749506238, −5.05171863136366667819298307792, −4.81532432350865930193125161278, −4.46290480656056156750578325378, −4.38254680231810686805080187457, −4.27180711269304897947265671847, −3.85565269632063985398092528753, −3.77576094433654918868844930838, −3.48353639321229734455327677952, −3.46503508504033697673268738029, −2.83884163364883867871465010789, −2.81792248649561282165864562735, −2.73213947864518013787378141391, −2.63434620837729844268528326940, −1.93184028240536928334888804179, −1.66357846160614310160657713703, −0.952230693505777515026116116314, −0.846955715637141101927702635843, −0.51147160771412702295183990121, −0.28521836138897651654208309530,
0.28521836138897651654208309530, 0.51147160771412702295183990121, 0.846955715637141101927702635843, 0.952230693505777515026116116314, 1.66357846160614310160657713703, 1.93184028240536928334888804179, 2.63434620837729844268528326940, 2.73213947864518013787378141391, 2.81792248649561282165864562735, 2.83884163364883867871465010789, 3.46503508504033697673268738029, 3.48353639321229734455327677952, 3.77576094433654918868844930838, 3.85565269632063985398092528753, 4.27180711269304897947265671847, 4.38254680231810686805080187457, 4.46290480656056156750578325378, 4.81532432350865930193125161278, 5.05171863136366667819298307792, 5.05201035346064229935749506238, 5.25325373515817224139226357954, 5.32336525289576685220608852465, 5.91200830580157344034137548353, 5.98340164432540046319805181445, 6.28490646401135145631564581838