| L(s) = 1 | − 2·2-s − 4-s + 8·8-s + 10·11-s − 7·16-s − 20·22-s + 2·23-s − 25-s − 4·29-s − 14·32-s + 6·37-s + 12·43-s − 10·44-s − 4·46-s + 2·50-s − 16·53-s + 8·58-s + 35·64-s − 28·67-s − 14·71-s − 12·74-s − 20·79-s − 24·86-s + 80·88-s − 2·92-s + 100-s + 32·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s + 2.82·8-s + 3.01·11-s − 7/4·16-s − 4.26·22-s + 0.417·23-s − 1/5·25-s − 0.742·29-s − 2.47·32-s + 0.986·37-s + 1.82·43-s − 1.50·44-s − 0.589·46-s + 0.282·50-s − 2.19·53-s + 1.05·58-s + 35/8·64-s − 3.42·67-s − 1.66·71-s − 1.39·74-s − 2.25·79-s − 2.58·86-s + 8.52·88-s − 0.208·92-s + 1/10·100-s + 3.10·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−7.59315018372019842769294095239, −7.28342071225309798898505652310, −7.13167564560012152294416600100, −6.34215171511087571525667957813, −5.96735414787932054483964871891, −5.63752173484997695290532171122, −4.60087511347345218102319658198, −4.47191624657260288271626711721, −4.22351345808096997888083312699, −3.59632548449757607506915649618, −3.08850093557224289960924465680, −1.92610841189085090066739675799, −1.32164798123200269811823069632, −1.12292208190814267208777122415, 0,
1.12292208190814267208777122415, 1.32164798123200269811823069632, 1.92610841189085090066739675799, 3.08850093557224289960924465680, 3.59632548449757607506915649618, 4.22351345808096997888083312699, 4.47191624657260288271626711721, 4.60087511347345218102319658198, 5.63752173484997695290532171122, 5.96735414787932054483964871891, 6.34215171511087571525667957813, 7.13167564560012152294416600100, 7.28342071225309798898505652310, 7.59315018372019842769294095239
