| L(s) = 1 | − 20·7-s − 9·9-s + 46·25-s − 76·31-s + 202·49-s + 180·63-s − 100·73-s + 116·79-s + 81·81-s − 380·97-s − 20·103-s + 142·121-s + ⋯ |
| L(s) = 1 | − 2.85·7-s − 9-s + 1.83·25-s − 2.45·31-s + 4.12·49-s + 20/7·63-s − 1.36·73-s + 1.46·79-s + 81-s − 3.91·97-s − 0.194·103-s + 1.17·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01886737810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01886737810\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 142 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 818 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 38 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3218 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6862 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4178 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 190 T + p^{2} T^{2} )^{2} \) |
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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
−10.43035248557573640235050703738, −9.767007081464545290530271006547, −9.464146010356835504146729453456, −9.113794001124889926639126798960, −8.831620550277877896122846696761, −8.364568472834463303685792094815, −7.63398212579587543106380995877, −7.15742536086699550173109596587, −6.80503639046343421827570143715, −6.37148167477301171479241778535, −6.09020641133191062633505379240, −5.41976122649572809557997901164, −5.22398655645033158717428551191, −4.27028590879526056818359667723, −3.56916677160683329053840820316, −3.41780303743324726664567926414, −2.78468430053239602587745237198, −2.44160318751001803912962730972, −1.20395447230219946815278538827, −0.04994528787496934983933144159,
0.04994528787496934983933144159, 1.20395447230219946815278538827, 2.44160318751001803912962730972, 2.78468430053239602587745237198, 3.41780303743324726664567926414, 3.56916677160683329053840820316, 4.27028590879526056818359667723, 5.22398655645033158717428551191, 5.41976122649572809557997901164, 6.09020641133191062633505379240, 6.37148167477301171479241778535, 6.80503639046343421827570143715, 7.15742536086699550173109596587, 7.63398212579587543106380995877, 8.364568472834463303685792094815, 8.831620550277877896122846696761, 9.113794001124889926639126798960, 9.464146010356835504146729453456, 9.767007081464545290530271006547, 10.43035248557573640235050703738
