L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s + 4·17-s − 8·19-s − 6·25-s − 4·27-s − 16·33-s − 12·41-s + 8·43-s − 14·49-s − 8·51-s + 16·57-s + 8·59-s − 8·67-s + 20·73-s + 12·75-s + 5·81-s − 8·83-s − 12·89-s + 4·97-s + 24·99-s − 24·107-s + 36·113-s + 26·121-s + 24·123-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s + 0.970·17-s − 1.83·19-s − 6/5·25-s − 0.769·27-s − 2.78·33-s − 1.87·41-s + 1.21·43-s − 2·49-s − 1.12·51-s + 2.11·57-s + 1.04·59-s − 0.977·67-s + 2.34·73-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s + 0.406·97-s + 2.41·99-s − 2.32·107-s + 3.38·113-s + 2.36·121-s + 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6426446320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6426446320\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.31725686074733747032880351584, −11.58204746144152243175303945055, −11.45788576598466513578332775969, −10.68009321996461187722201460512, −9.964671915727173343349228861429, −9.549120050570062839357860330869, −8.777512613188475551383845278968, −8.098990694093691505068092710868, −7.13146007277855219299925953915, −6.42897107072744719896577758454, −6.21040581024747445781537308376, −5.24179959934618595033608799542, −4.25303028692796488061253338187, −3.71523857372513194753595622108, −1.64229196717434907696557687175,
1.64229196717434907696557687175, 3.71523857372513194753595622108, 4.25303028692796488061253338187, 5.24179959934618595033608799542, 6.21040581024747445781537308376, 6.42897107072744719896577758454, 7.13146007277855219299925953915, 8.098990694093691505068092710868, 8.777512613188475551383845278968, 9.549120050570062839357860330869, 9.964671915727173343349228861429, 10.68009321996461187722201460512, 11.45788576598466513578332775969, 11.58204746144152243175303945055, 12.31725686074733747032880351584