L(s) = 1 | − 12·13-s − 10·25-s + 4·31-s + 12·43-s + 28·49-s + 28·79-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 58·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | − 3.32·13-s − 2·25-s + 0.718·31-s + 1.82·43-s + 4·49-s + 3.15·79-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.735805351\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.735805351\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 26 T^{2} + 387 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 38 T^{2} + 603 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 34 T^{2} - 1053 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.87845315392843702853516523045, −5.77265204218054955160341709214, −5.44893240258135787977669476031, −5.18257772447387315555101796686, −5.17558962941484546918972587441, −5.01758964152144756014916933690, −4.70163008242966997552776427781, −4.67754535478360926184003696163, −4.25436441247909815898468352107, −4.07191867146720755965702826899, −3.97364296819891083516776269075, −3.69828980810626040488051729426, −3.69485953801778671907384499079, −3.25607780186457010780812443178, −2.77243438767779922137625155070, −2.75746782913722507625549423488, −2.48792468443495696564972397714, −2.35368336687255109312609947029, −2.28960945413365497212370165803, −1.96225899038680630751959293581, −1.59389906125068671869749888824, −1.28706347852677090172308069466, −0.76845076939302708249878810088, −0.61439870339617032009455323443, −0.23650946776816337623402015927,
0.23650946776816337623402015927, 0.61439870339617032009455323443, 0.76845076939302708249878810088, 1.28706347852677090172308069466, 1.59389906125068671869749888824, 1.96225899038680630751959293581, 2.28960945413365497212370165803, 2.35368336687255109312609947029, 2.48792468443495696564972397714, 2.75746782913722507625549423488, 2.77243438767779922137625155070, 3.25607780186457010780812443178, 3.69485953801778671907384499079, 3.69828980810626040488051729426, 3.97364296819891083516776269075, 4.07191867146720755965702826899, 4.25436441247909815898468352107, 4.67754535478360926184003696163, 4.70163008242966997552776427781, 5.01758964152144756014916933690, 5.17558962941484546918972587441, 5.18257772447387315555101796686, 5.44893240258135787977669476031, 5.77265204218054955160341709214, 5.87845315392843702853516523045