| L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s − 4·13-s − 8·17-s + 4·23-s + 2·25-s − 4·27-s + 4·29-s + 8·31-s − 8·33-s − 4·37-s + 8·39-s − 16·41-s − 16·43-s − 8·47-s + 16·51-s − 4·53-s + 16·59-s + 4·61-s + 8·67-s − 8·69-s + 12·71-s − 12·73-s − 4·75-s − 8·79-s + 5·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s − 1.10·13-s − 1.94·17-s + 0.834·23-s + 2/5·25-s − 0.769·27-s + 0.742·29-s + 1.43·31-s − 1.39·33-s − 0.657·37-s + 1.28·39-s − 2.49·41-s − 2.43·43-s − 1.16·47-s + 2.24·51-s − 0.549·53-s + 2.08·59-s + 0.512·61-s + 0.977·67-s − 0.963·69-s + 1.42·71-s − 1.40·73-s − 0.461·75-s − 0.900·79-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22127616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8903568948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8903568948\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.603924280029396216137427429141, −8.270091430142331373694486550974, −7.62327723976503701743025320193, −7.20966357404561424607910777523, −6.81988881294969734229411687633, −6.61144376108141267212189361095, −6.36335560271126072953218819983, −6.32238596233301334453419553162, −5.21465422444213022628059632554, −5.12936950895129047366890772971, −4.94314398663802256829024930968, −4.61486582547975071262776718261, −3.88934075823611853422117487540, −3.82710586416118319098629080857, −3.04441679227922769430397884341, −2.72165894477926099189692458740, −1.86137984169800998788670805297, −1.79025268275556139524750129682, −0.970771569109480339621097253502, −0.32593782165191865726222136075,
0.32593782165191865726222136075, 0.970771569109480339621097253502, 1.79025268275556139524750129682, 1.86137984169800998788670805297, 2.72165894477926099189692458740, 3.04441679227922769430397884341, 3.82710586416118319098629080857, 3.88934075823611853422117487540, 4.61486582547975071262776718261, 4.94314398663802256829024930968, 5.12936950895129047366890772971, 5.21465422444213022628059632554, 6.32238596233301334453419553162, 6.36335560271126072953218819983, 6.61144376108141267212189361095, 6.81988881294969734229411687633, 7.20966357404561424607910777523, 7.62327723976503701743025320193, 8.270091430142331373694486550974, 8.603924280029396216137427429141
