L(s) = 1 | + 10·9-s − 12·17-s − 4·25-s − 22·49-s − 4·73-s + 57·81-s + 72·89-s − 56·97-s + 24·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 120·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 10/3·9-s − 2.91·17-s − 4/5·25-s − 3.14·49-s − 0.468·73-s + 19/3·81-s + 7.63·89-s − 5.68·97-s + 2.25·113-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 9.70·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.107963332\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.107963332\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 79 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
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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
−6.84619942592678888942919242058, −6.83927656353014488929483641243, −6.77831619077831555211479370128, −6.22929062557113963510343636474, −6.04354071914397119737973510857, −6.01880978994877518340702791784, −5.90208490320469562409586990325, −5.08185145654037014623175533490, −5.04860448240018613682534738912, −4.81557542890447627150786534131, −4.62019438664337834838522489439, −4.56267758728510822804505539546, −4.29510467888817761876833666905, −3.92553215443835919497526020369, −3.78146532603560973958805244249, −3.63761215572347329031137936426, −3.10761673108875031430359833063, −3.01868337391652168154813455937, −2.45550167055819547236929695384, −1.99570936840885491694436235401, −1.99038635616161493495047140663, −1.66640161108249803549929747609, −1.57573625160565060131243997498, −0.73052321173957581962420246312, −0.52327539071794820116424229984,
0.52327539071794820116424229984, 0.73052321173957581962420246312, 1.57573625160565060131243997498, 1.66640161108249803549929747609, 1.99038635616161493495047140663, 1.99570936840885491694436235401, 2.45550167055819547236929695384, 3.01868337391652168154813455937, 3.10761673108875031430359833063, 3.63761215572347329031137936426, 3.78146532603560973958805244249, 3.92553215443835919497526020369, 4.29510467888817761876833666905, 4.56267758728510822804505539546, 4.62019438664337834838522489439, 4.81557542890447627150786534131, 5.04860448240018613682534738912, 5.08185145654037014623175533490, 5.90208490320469562409586990325, 6.01880978994877518340702791784, 6.04354071914397119737973510857, 6.22929062557113963510343636474, 6.77831619077831555211479370128, 6.83927656353014488929483641243, 6.84619942592678888942919242058