| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s + 7·13-s + 16-s + 6·22-s − 3·23-s + 2·25-s − 7·26-s − 32-s − 17·37-s − 6·44-s + 3·46-s − 6·47-s − 4·49-s − 2·50-s + 7·52-s − 3·59-s − 5·61-s + 64-s + 9·71-s − 8·73-s + 17·74-s − 6·83-s + 6·88-s − 3·92-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1.94·13-s + 1/4·16-s + 1.27·22-s − 0.625·23-s + 2/5·25-s − 1.37·26-s − 0.176·32-s − 2.79·37-s − 0.904·44-s + 0.442·46-s − 0.875·47-s − 4/7·49-s − 0.282·50-s + 0.970·52-s − 0.390·59-s − 0.640·61-s + 1/8·64-s + 1.06·71-s − 0.936·73-s + 1.97·74-s − 0.658·83-s + 0.639·88-s − 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−8.629385276507449803961914456874, −8.441118034331777184341958900492, −8.024388843455897226029319866828, −7.51444591180234071216773030473, −6.96460162984252591258296745907, −6.45406965081292175877524735543, −5.95034013745279661508505530031, −5.40095774134454794341597659061, −5.00038616437170600269460847264, −4.15833731826171789779502059600, −3.40222320204255249264985227549, −3.04368076244806356394642727093, −2.09352024097234021857032306379, −1.39358619364689210941072500297, 0,
1.39358619364689210941072500297, 2.09352024097234021857032306379, 3.04368076244806356394642727093, 3.40222320204255249264985227549, 4.15833731826171789779502059600, 5.00038616437170600269460847264, 5.40095774134454794341597659061, 5.95034013745279661508505530031, 6.45406965081292175877524735543, 6.96460162984252591258296745907, 7.51444591180234071216773030473, 8.024388843455897226029319866828, 8.441118034331777184341958900492, 8.629385276507449803961914456874
