| L(s) = 1 | + 4·5-s + 2·13-s + 6·17-s + 11·25-s + 14·37-s − 16·41-s − 18·53-s − 24·61-s + 8·65-s − 22·73-s − 9·81-s + 24·85-s + 26·97-s − 4·101-s − 2·113-s + 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.554·13-s + 1.45·17-s + 11/5·25-s + 2.30·37-s − 2.49·41-s − 2.47·53-s − 3.07·61-s + 0.992·65-s − 2.57·73-s − 81-s + 2.60·85-s + 2.63·97-s − 0.398·101-s − 0.188·113-s + 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298856930\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−11.67884295715023212621534687179, −11.55357618751306773912596348696, −10.72213307634961733674798426066, −10.42821154254018852586134744406, −9.938012052388440123704117322938, −9.693419260056799415960468756177, −9.029930109482513194692993201510, −8.859010812383679043032619910844, −7.85329864362359783038199561841, −7.83780976762986300410096230967, −6.88035734678523391819824097259, −6.40400267991988480893327881443, −5.80780206704734417757645076831, −5.75800316685793390864568868433, −4.84141601387313376758748006505, −4.46793970751731429499311987066, −3.15954120480596599270267330916, −3.07029287086804376795319408489, −1.85772778075380890368705227763, −1.32586343342437555983892467440,
1.32586343342437555983892467440, 1.85772778075380890368705227763, 3.07029287086804376795319408489, 3.15954120480596599270267330916, 4.46793970751731429499311987066, 4.84141601387313376758748006505, 5.75800316685793390864568868433, 5.80780206704734417757645076831, 6.40400267991988480893327881443, 6.88035734678523391819824097259, 7.83780976762986300410096230967, 7.85329864362359783038199561841, 8.859010812383679043032619910844, 9.029930109482513194692993201510, 9.693419260056799415960468756177, 9.938012052388440123704117322938, 10.42821154254018852586134744406, 10.72213307634961733674798426066, 11.55357618751306773912596348696, 11.67884295715023212621534687179
