L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 9-s + 14-s + 16-s − 18-s − 4·19-s − 2·25-s − 28-s − 4·29-s − 16·31-s − 32-s + 36-s + 4·38-s − 4·47-s + 49-s + 2·50-s − 8·53-s + 56-s + 4·58-s + 8·59-s + 16·62-s − 63-s + 64-s − 72-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 2/5·25-s − 0.188·28-s − 0.742·29-s − 2.87·31-s − 0.176·32-s + 1/6·36-s + 0.648·38-s − 0.583·47-s + 1/7·49-s + 0.282·50-s − 1.09·53-s + 0.133·56-s + 0.525·58-s + 1.04·59-s + 2.03·62-s − 0.125·63-s + 1/8·64-s − 0.117·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
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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
−9.414083273461708020163754846234, −8.850834693875306878942371642332, −8.467404021459094466150707986452, −7.84402133530442139109035582825, −7.30660152168727094665841643577, −7.00366424188441461862683629447, −6.36913756339708829690555340663, −5.77470604093417035010896820852, −5.34672214305477453710242611042, −4.46773279343609440243528731656, −3.80679823357620379763116820105, −3.26528166288566083293814995391, −2.24368686488033637177806314655, −1.61703632224706865588294405099, 0,
1.61703632224706865588294405099, 2.24368686488033637177806314655, 3.26528166288566083293814995391, 3.80679823357620379763116820105, 4.46773279343609440243528731656, 5.34672214305477453710242611042, 5.77470604093417035010896820852, 6.36913756339708829690555340663, 7.00366424188441461862683629447, 7.30660152168727094665841643577, 7.84402133530442139109035582825, 8.467404021459094466150707986452, 8.850834693875306878942371642332, 9.414083273461708020163754846234