| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s − 9-s − 8·14-s − 4·16-s + 4·17-s − 2·18-s − 25-s − 8·28-s − 12·31-s − 8·32-s + 8·34-s − 2·36-s + 4·41-s − 2·49-s − 2·50-s − 24·62-s + 4·63-s − 8·64-s + 8·68-s − 16·71-s + 20·73-s + 12·79-s + 81-s + 8·82-s + 12·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s − 2.13·14-s − 16-s + 0.970·17-s − 0.471·18-s − 1/5·25-s − 1.51·28-s − 2.15·31-s − 1.41·32-s + 1.37·34-s − 1/3·36-s + 0.624·41-s − 2/7·49-s − 0.282·50-s − 3.04·62-s + 0.503·63-s − 64-s + 0.970·68-s − 1.89·71-s + 2.34·73-s + 1.35·79-s + 1/9·81-s + 0.883·82-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.215306167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215306167\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.638471008516639167629170350671, −9.618073485672695632543889140831, −9.017785275926625322074069984505, −8.842093072953548982634440196592, −8.096153394494681126626379979479, −7.63258669140183318051372938942, −7.33815811557229557774346678912, −6.59872974742946963904696170901, −6.57728991501454290809645543432, −5.91126596819999716178306450491, −5.77418716619262750038753553825, −5.08074714770612253828345495879, −4.99004399972470679869031049815, −3.99047038824296281586775739118, −3.87027941896783486989026666301, −3.25006794480691946776905133526, −3.07390542123489520181533913969, −2.37433466960853583352494248748, −1.71109911464624447268624682019, −0.46624707050868904609824847920,
0.46624707050868904609824847920, 1.71109911464624447268624682019, 2.37433466960853583352494248748, 3.07390542123489520181533913969, 3.25006794480691946776905133526, 3.87027941896783486989026666301, 3.99047038824296281586775739118, 4.99004399972470679869031049815, 5.08074714770612253828345495879, 5.77418716619262750038753553825, 5.91126596819999716178306450491, 6.57728991501454290809645543432, 6.59872974742946963904696170901, 7.33815811557229557774346678912, 7.63258669140183318051372938942, 8.096153394494681126626379979479, 8.842093072953548982634440196592, 9.017785275926625322074069984505, 9.618073485672695632543889140831, 9.638471008516639167629170350671
