| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s − 4·13-s + 4·15-s − 25-s − 4·27-s + 16·31-s + 4·37-s + 8·39-s + 12·41-s + 8·43-s − 6·45-s + 10·49-s − 12·53-s + 8·65-s − 8·71-s + 2·75-s − 16·79-s + 5·81-s − 24·83-s + 28·89-s − 32·93-s − 24·107-s − 8·111-s − 12·117-s − 14·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s + 2.87·31-s + 0.657·37-s + 1.28·39-s + 1.87·41-s + 1.21·43-s − 0.894·45-s + 10/7·49-s − 1.64·53-s + 0.992·65-s − 0.949·71-s + 0.230·75-s − 1.80·79-s + 5/9·81-s − 2.63·83-s + 2.96·89-s − 3.31·93-s − 2.32·107-s − 0.759·111-s − 1.10·117-s − 1.27·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8243148612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8243148612\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.670847643029248028092444877962, −8.165804499750410969413228790240, −7.80427070371199971056721954318, −7.49413924637445370228607127530, −7.34038455312959905105964029282, −6.85949254509761650764275651052, −6.28905735943566354378836079933, −6.11428579330236840274088977616, −5.85994088413964385767583043704, −5.14936883304616687481552464239, −4.97857079798671250960610203551, −4.43301736116894131327615038362, −4.11931217473419772273132318954, −4.04949561136821546815550243144, −3.12822481043466951292579718225, −2.56799039311893495783023479249, −2.53957795830406123174291566226, −1.47394261365309494293767836864, −0.967272630754393992379054460501, −0.35898832583721480186938109754,
0.35898832583721480186938109754, 0.967272630754393992379054460501, 1.47394261365309494293767836864, 2.53957795830406123174291566226, 2.56799039311893495783023479249, 3.12822481043466951292579718225, 4.04949561136821546815550243144, 4.11931217473419772273132318954, 4.43301736116894131327615038362, 4.97857079798671250960610203551, 5.14936883304616687481552464239, 5.85994088413964385767583043704, 6.11428579330236840274088977616, 6.28905735943566354378836079933, 6.85949254509761650764275651052, 7.34038455312959905105964029282, 7.49413924637445370228607127530, 7.80427070371199971056721954318, 8.165804499750410969413228790240, 8.670847643029248028092444877962
