| L(s) = 1 | − 2·9-s − 12·17-s + 12·41-s − 28·49-s + 4·73-s + 9·81-s − 36·89-s − 40·97-s + 36·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 2.91·17-s + 1.87·41-s − 4·49-s + 0.468·73-s + 81-s − 3.81·89-s − 4.06·97-s + 3.38·113-s + 1.27·121-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 + 1.94·153-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 + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610357683\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.53455498020109620982278037090, −6.51455866753393542790214056899, −6.37979273506460949545321030173, −6.09127159967759435054672510756, −5.93995314542723740075259388663, −5.61092136861478168337222489920, −5.45699583734831101347793920249, −5.08579881745749294888630438122, −4.90898720589016166637762190945, −4.84258022410827206373677875929, −4.42653700757017734927518446000, −4.19537219932644720003695052693, −4.08664443615262597745045310805, −4.02655996405017393432138166371, −3.40729810453939568398121871495, −3.30173201117611897198000901397, −2.86483721348894094557734912271, −2.71185581064950569311684051030, −2.65819863299756873929878575094, −2.11322504685344925314654835613, −1.91519532285394232707928963630, −1.56787672600461043390312226791, −1.37891358579616000913528935289, −0.46999951498999245449660171786, −0.40141296731276892723743300050,
0.40141296731276892723743300050, 0.46999951498999245449660171786, 1.37891358579616000913528935289, 1.56787672600461043390312226791, 1.91519532285394232707928963630, 2.11322504685344925314654835613, 2.65819863299756873929878575094, 2.71185581064950569311684051030, 2.86483721348894094557734912271, 3.30173201117611897198000901397, 3.40729810453939568398121871495, 4.02655996405017393432138166371, 4.08664443615262597745045310805, 4.19537219932644720003695052693, 4.42653700757017734927518446000, 4.84258022410827206373677875929, 4.90898720589016166637762190945, 5.08579881745749294888630438122, 5.45699583734831101347793920249, 5.61092136861478168337222489920, 5.93995314542723740075259388663, 6.09127159967759435054672510756, 6.37979273506460949545321030173, 6.51455866753393542790214056899, 6.53455498020109620982278037090
