| L(s) = 1 | + 128·2-s + 8.19e3·4-s − 6.76e4·5-s − 8.65e6·10-s + 2.67e7·13-s − 6.71e7·16-s − 1.90e8·17-s − 5.53e8·20-s + 3.34e9·25-s + 3.42e9·26-s − 8.58e9·32-s − 2.43e10·34-s − 9.17e9·37-s − 7.39e10·41-s + 4.28e11·50-s + 2.19e11·52-s − 4.34e11·53-s − 1.48e12·61-s − 5.49e11·64-s − 1.81e12·65-s − 1.55e12·68-s + 2.70e12·73-s − 1.17e12·74-s + 4.53e12·80-s − 2.54e12·81-s − 9.46e12·82-s + 1.28e13·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.93·5-s − 2.73·10-s + 1.53·13-s − 16-s − 1.91·17-s − 1.93·20-s + 2.74·25-s + 2.17·26-s − 1.41·32-s − 2.70·34-s − 0.587·37-s − 2.43·41-s + 3.88·50-s + 1.53·52-s − 2.69·53-s − 3.69·61-s − 64-s − 2.97·65-s − 1.91·68-s + 2.09·73-s − 0.830·74-s + 1.93·80-s − 81-s − 3.43·82-s + 3.69·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.278318951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.278318951\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_2$ | \( 1 - p^{7} T + p^{13} T^{2} \) |
| 5 | $C_2$ | \( 1 + 67604 T + p^{13} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 34042324 T + p^{13} T^{2} )( 1 + 7254006 T + p^{13} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8582078 T + p^{13} T^{2} )( 1 + 198858392 T + p^{13} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1486672730 T + p^{13} T^{2} )( 1 + 1486672730 T + p^{13} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17004695748 T + p^{13} T^{2} )( 1 + 26174743922 T + p^{13} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 36966257128 T + p^{13} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 147965133676 T + p^{13} T^{2} )( 1 + 286798198946 T + p^{13} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 743365892868 T + p^{13} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{13} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2582975771994 T + p^{13} T^{2} )( 1 - 125036758064 T + p^{13} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{13} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{26} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7780304165930 T + p^{13} T^{2} )( 1 + 7780304165930 T + p^{13} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10874684920338 T + p^{13} T^{2} )( 1 + 12286255082392 T + p^{13} T^{2} ) \) |
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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
−15.28168368501672427278328048232, −15.26721704030479583985654868952, −14.00323857959254384435403288635, −13.67510382243832145766313462380, −12.78585728371268309097964436348, −12.41334678543929727564826836872, −11.39119548888974211566476704253, −11.36210494483222552073066695319, −10.60097142339016742784497004755, −8.966198576789825979854047873049, −8.592092234690978076508633430392, −7.67672212420007540558223929409, −6.70425696464635230986061494871, −6.23792111844537285106983128884, −4.80150228676562951900302706688, −4.49406762528373942429351796975, −3.50578535237240952109499222189, −3.22482021299549802839596884494, −1.74450751169536126594232161792, −0.31110364076559724148041411240,
0.31110364076559724148041411240, 1.74450751169536126594232161792, 3.22482021299549802839596884494, 3.50578535237240952109499222189, 4.49406762528373942429351796975, 4.80150228676562951900302706688, 6.23792111844537285106983128884, 6.70425696464635230986061494871, 7.67672212420007540558223929409, 8.592092234690978076508633430392, 8.966198576789825979854047873049, 10.60097142339016742784497004755, 11.36210494483222552073066695319, 11.39119548888974211566476704253, 12.41334678543929727564826836872, 12.78585728371268309097964436348, 13.67510382243832145766313462380, 14.00323857959254384435403288635, 15.26721704030479583985654868952, 15.28168368501672427278328048232
