| L(s) = 1 | + 16·9-s + 48·19-s + 128·31-s − 128·49-s − 64·61-s − 288·79-s + 175·81-s − 320·109-s − 60·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 768·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 16/9·9-s + 2.52·19-s + 4.12·31-s − 2.61·49-s − 1.04·61-s − 3.64·79-s + 2.16·81-s − 2.93·109-s − 0.495·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 4.49·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6112432056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6112432056\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 16 T^{2} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 450 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 12 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 480 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 2194 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 2032 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3168 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6690 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8944 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2942 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 11856 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 11490 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 7838 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.61184066140171455473676388012, −6.48709551863950520439863577165, −6.48581659268062826958060133545, −6.16109198759364986931253941388, −5.90693935801490969306294248220, −5.79659448464485781898890150901, −5.16765947784305652682658031117, −5.07602356160907144401755283967, −5.06350918940668859887237603715, −4.78879221773003813561571662231, −4.44690160496100794466031510392, −4.18987227407716151068085793714, −4.17917065910469650127690282118, −3.79089692552468597083104527066, −3.29980427480520679415704628265, −3.27034247340394124367875606633, −2.88370427983083867134101511786, −2.85006919593821218080080763369, −2.28563088645017281250323724353, −2.19290996970596016459492930421, −1.42218358211956414483178451138, −1.28808868642069214270100484425, −1.16540125918190378514255367288, −1.00292540316474624783634881548, −0.092957882867600831105128195902,
0.092957882867600831105128195902, 1.00292540316474624783634881548, 1.16540125918190378514255367288, 1.28808868642069214270100484425, 1.42218358211956414483178451138, 2.19290996970596016459492930421, 2.28563088645017281250323724353, 2.85006919593821218080080763369, 2.88370427983083867134101511786, 3.27034247340394124367875606633, 3.29980427480520679415704628265, 3.79089692552468597083104527066, 4.17917065910469650127690282118, 4.18987227407716151068085793714, 4.44690160496100794466031510392, 4.78879221773003813561571662231, 5.06350918940668859887237603715, 5.07602356160907144401755283967, 5.16765947784305652682658031117, 5.79659448464485781898890150901, 5.90693935801490969306294248220, 6.16109198759364986931253941388, 6.48581659268062826958060133545, 6.48709551863950520439863577165, 6.61184066140171455473676388012
