| L(s) = 1 | − 2·2-s + 2·4-s − 4·16-s + 25-s + 4·29-s + 8·32-s − 18·37-s − 7·49-s − 2·50-s + 8·53-s − 8·58-s − 8·64-s + 36·74-s + 14·98-s + 2·100-s − 16·106-s + 18·109-s + 20·113-s + 8·116-s − 18·121-s + 127-s + 131-s + 137-s + 139-s − 36·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 16-s + 1/5·25-s + 0.742·29-s + 1.41·32-s − 2.95·37-s − 49-s − 0.282·50-s + 1.09·53-s − 1.05·58-s − 64-s + 4.18·74-s + 1.41·98-s + 1/5·100-s − 1.55·106-s + 1.72·109-s + 1.88·113-s + 0.742·116-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.95·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−8.466017411901143198132464323899, −7.82405911296548350465296941916, −7.39833327293981923265521718612, −7.00838209199160818860423185084, −6.61792132221494890316874392625, −6.11952612430237780592522941075, −5.42626844519663349852402098884, −4.94872428076579067656314526985, −4.50323772228931333330026823440, −3.71991151392418890184811040035, −3.23587240730897733024119524678, −2.40764058755436774779739927582, −1.81774528928970206867476848200, −1.08036623521030087240450162057, 0,
1.08036623521030087240450162057, 1.81774528928970206867476848200, 2.40764058755436774779739927582, 3.23587240730897733024119524678, 3.71991151392418890184811040035, 4.50323772228931333330026823440, 4.94872428076579067656314526985, 5.42626844519663349852402098884, 6.11952612430237780592522941075, 6.61792132221494890316874392625, 7.00838209199160818860423185084, 7.39833327293981923265521718612, 7.82405911296548350465296941916, 8.466017411901143198132464323899
