| L(s) = 1 | + 4-s − 25-s + 2·37-s − 4·43-s − 64-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s − 25-s + 2·37-s − 4·43-s − 64-s − 2·67-s − 2·79-s − 100-s + 2·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8637825753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8637825753\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−11.49438970359066473817352013659, −11.33626292852082760209850477605, −10.67421072164152299370820510375, −10.26463573503206159132979439772, −9.698700915356889802984301980440, −9.609722008000977215470084698991, −8.591224729354076119968779401602, −8.549787287379746596472714024378, −7.78623839517945642711281070580, −7.45134713216296716150282292034, −6.87834066262158608905266997488, −6.52676742060248642637602831359, −5.90646054431188370473342496616, −5.63467893600201756616122681466, −4.61122384632602001394216453963, −4.48029662965645927091740316497, −3.39923831260526179066932882330, −3.05249497320041956745119721480, −2.16617937203495717213378069952, −1.60038641080423647027116313974,
1.60038641080423647027116313974, 2.16617937203495717213378069952, 3.05249497320041956745119721480, 3.39923831260526179066932882330, 4.48029662965645927091740316497, 4.61122384632602001394216453963, 5.63467893600201756616122681466, 5.90646054431188370473342496616, 6.52676742060248642637602831359, 6.87834066262158608905266997488, 7.45134713216296716150282292034, 7.78623839517945642711281070580, 8.549787287379746596472714024378, 8.591224729354076119968779401602, 9.609722008000977215470084698991, 9.698700915356889802984301980440, 10.26463573503206159132979439772, 10.67421072164152299370820510375, 11.33626292852082760209850477605, 11.49438970359066473817352013659
