L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s + 2·9-s + 4·10-s − 16-s − 2·18-s + 4·20-s + 2·25-s − 5·32-s − 2·36-s − 4·37-s − 12·40-s − 6·41-s − 8·45-s − 6·49-s − 2·50-s − 4·61-s + 7·64-s + 6·72-s − 28·73-s + 4·74-s + 4·80-s − 5·81-s + 6·82-s + 8·90-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 2/3·9-s + 1.26·10-s − 1/4·16-s − 0.471·18-s + 0.894·20-s + 2/5·25-s − 0.883·32-s − 1/3·36-s − 0.657·37-s − 1.89·40-s − 0.937·41-s − 1.19·45-s − 6/7·49-s − 0.282·50-s − 0.512·61-s + 7/8·64-s + 0.707·72-s − 3.27·73-s + 0.464·74-s + 0.447·80-s − 5/9·81-s + 0.662·82-s + 0.843·90-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 41 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−10.19023449659800280025304267480, −9.950081302442658861535765863833, −9.136661245630303130527527106950, −8.751287964350617633590160637874, −8.062173761857703697875994376929, −7.85645423889849417776793402919, −7.25560493967332643536479769551, −6.83969790329135228911236082140, −5.83159640611586890531957946716, −4.97376859736062822975389284623, −4.33835034948233865658543563854, −3.93725321871172263777531833941, −3.18023855880530203519615252116, −1.58743281801218509896114757290, 0,
1.58743281801218509896114757290, 3.18023855880530203519615252116, 3.93725321871172263777531833941, 4.33835034948233865658543563854, 4.97376859736062822975389284623, 5.83159640611586890531957946716, 6.83969790329135228911236082140, 7.25560493967332643536479769551, 7.85645423889849417776793402919, 8.062173761857703697875994376929, 8.751287964350617633590160637874, 9.136661245630303130527527106950, 9.950081302442658861535765863833, 10.19023449659800280025304267480