L(s) = 1 | − 31·5-s + 73·7-s + 162·9-s + 233·11-s + 353·17-s + 722·19-s − 316·23-s + 625·25-s − 2.26e3·35-s − 3.52e3·43-s − 5.02e3·45-s − 1.20e3·47-s + 2.40e3·49-s − 7.22e3·55-s − 3.16e3·61-s + 1.18e4·63-s + 1.00e4·73-s + 1.70e4·77-s + 1.96e4·81-s − 1.13e4·83-s − 1.09e4·85-s − 2.23e4·95-s + 3.77e4·99-s − 1.99e4·101-s + 9.79e3·115-s + 2.57e4·119-s + 1.46e4·121-s + ⋯ |
L(s) = 1 | − 1.23·5-s + 1.48·7-s + 2·9-s + 1.92·11-s + 1.22·17-s + 2·19-s − 0.597·23-s + 25-s − 1.84·35-s − 1.90·43-s − 2.47·45-s − 0.546·47-s + 49-s − 2.38·55-s − 0.851·61-s + 2.97·63-s + 1.88·73-s + 2.86·77-s + 3·81-s − 1.64·83-s − 1.51·85-s − 2.47·95-s + 3.85·99-s − 1.96·101-s + 0.740·115-s + 1.81·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.157129682\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.157129682\) |
\(L(3)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 31 T + 336 T^{2} + 31 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 73 T + 2928 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 233 T + 39648 T^{2} - 233 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 353 T + 41088 T^{2} - 353 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 158 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3527 T + 9020928 T^{2} + 3527 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1207 T - 3422832 T^{2} + 1207 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 5678 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−14.01904700270578621902468101096, −13.74744562895246542081352818785, −12.74095326069789419674494380532, −12.21847304437388788078895394684, −11.70864465853710102278416160415, −11.64468282297172972930385943609, −10.85774403611172679270625016559, −9.981556671109526718291112115944, −9.682769044989625067952066346482, −8.918084673256899788358037017479, −7.894755410937480218167634326307, −7.87035126533830974706040219243, −7.09606906488378120870053667561, −6.60156746884170893242163577892, −5.29732262375819872334377315942, −4.67789452048106077389420418400, −3.96023389592170013474671415225, −3.49104535304896258821772151362, −1.42783578818619358528327281108, −1.22965593474006071425227239528,
1.22965593474006071425227239528, 1.42783578818619358528327281108, 3.49104535304896258821772151362, 3.96023389592170013474671415225, 4.67789452048106077389420418400, 5.29732262375819872334377315942, 6.60156746884170893242163577892, 7.09606906488378120870053667561, 7.87035126533830974706040219243, 7.894755410937480218167634326307, 8.918084673256899788358037017479, 9.682769044989625067952066346482, 9.981556671109526718291112115944, 10.85774403611172679270625016559, 11.64468282297172972930385943609, 11.70864465853710102278416160415, 12.21847304437388788078895394684, 12.74095326069789419674494380532, 13.74744562895246542081352818785, 14.01904700270578621902468101096