| L(s) = 1 | + 3·4-s + 5·16-s + 20·25-s − 24·37-s − 14·49-s + 3·64-s + 60·100-s + 72·109-s + 12·121-s + 127-s + 131-s + 137-s + 139-s − 72·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 42·196-s + 197-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5/4·16-s + 4·25-s − 3.94·37-s − 2·49-s + 3/8·64-s + 6·100-s + 6.89·109-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.91·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 3·196-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.716458415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.716458415\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−8.783625229935746385680216189533, −8.468395675941094319345143231234, −8.290023436584896162279437184041, −8.191166625525540201548625739406, −7.51514557866725144703633220496, −7.40818182939887828429104272260, −6.97417319036628040235627563948, −6.96018890078184126814115718819, −6.93731582991745748630645217553, −6.39791320212899000378501938355, −6.21588114120258823031094461626, −5.93147938552522879250393834542, −5.54748321384062432046459152552, −5.12278994592470895151430590929, −4.92719967959972289783433168782, −4.72726421005485716529762933742, −4.46865325691232563849339806941, −3.65096293991075743902802151125, −3.44127348092956702547536900357, −3.16590581121915030195496199756, −2.96645422190522964272183663463, −2.43370948788322615610044689888, −1.83279581159740924653256896651, −1.70012766171373320250956563777, −0.884868169770938756035335925143,
0.884868169770938756035335925143, 1.70012766171373320250956563777, 1.83279581159740924653256896651, 2.43370948788322615610044689888, 2.96645422190522964272183663463, 3.16590581121915030195496199756, 3.44127348092956702547536900357, 3.65096293991075743902802151125, 4.46865325691232563849339806941, 4.72726421005485716529762933742, 4.92719967959972289783433168782, 5.12278994592470895151430590929, 5.54748321384062432046459152552, 5.93147938552522879250393834542, 6.21588114120258823031094461626, 6.39791320212899000378501938355, 6.93731582991745748630645217553, 6.96018890078184126814115718819, 6.97417319036628040235627563948, 7.40818182939887828429104272260, 7.51514557866725144703633220496, 8.191166625525540201548625739406, 8.290023436584896162279437184041, 8.468395675941094319345143231234, 8.783625229935746385680216189533
