| L(s) = 1 | − 9.60e3·7-s + 1.79e4·13-s − 3.93e5·19-s − 5.14e6·25-s − 5.31e6·31-s − 5.08e6·37-s + 1.51e7·43-s + 5.76e7·49-s + 2.75e7·61-s + 1.19e8·67-s − 1.40e8·73-s + 3.39e8·79-s − 1.72e8·91-s + 7.46e8·97-s + 2.23e9·103-s + 1.67e9·109-s − 1.62e9·121-s + 127-s + 131-s + 3.77e9·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.174·13-s − 0.692·19-s − 2.63·25-s − 1.03·31-s − 0.446·37-s + 0.675·43-s + 10/7·49-s + 0.254·61-s + 0.726·67-s − 0.577·73-s + 0.979·79-s − 0.263·91-s + 0.855·97-s + 1.95·103-s + 1.13·109-s − 0.687·121-s + 1.04·133-s − 0.143·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.960832881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.960832881\) |
| \(L(\frac{11}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{4} T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 5147124 T^{2} + 530226988982 p^{2} T^{4} + 5147124 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 1621035756 T^{2} - 3316796889054303418 T^{4} + 1621035756 p^{18} T^{6} + p^{36} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 8988 T + 883456526 T^{2} - 8988 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 268718632164 T^{2} + \)\(41\!\cdots\!98\)\( T^{4} + 268718632164 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 196784 T + 33405823314 p T^{2} + 196784 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 4339319579004 T^{2} + \)\(95\!\cdots\!98\)\( T^{4} + 4339319579004 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 1655566388676 p T^{2} + \)\(97\!\cdots\!30\)\( T^{4} + 1655566388676 p^{19} T^{6} + p^{36} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2658096 T + 50067821800046 T^{2} + 2658096 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 2543252 T + 260543571099486 T^{2} + 2543252 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 51249345210500 T^{2} - \)\(16\!\cdots\!42\)\( T^{4} + 51249345210500 p^{18} T^{6} + p^{36} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 7573624 T + 764308228313766 T^{2} - 7573624 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 4265252924708156 T^{2} + \)\(70\!\cdots\!38\)\( T^{4} + 4265252924708156 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 10398364148258132 T^{2} + \)\(46\!\cdots\!34\)\( T^{4} + 10398364148258132 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 31718851834812012 T^{2} + \)\(40\!\cdots\!62\)\( T^{4} + 31718851834812012 p^{18} T^{6} + p^{36} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 13783476 T + 14438394138069326 T^{2} - 13783476 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 59894504 T + 55210212577486998 T^{2} - 59894504 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 89270113051354428 T^{2} + \)\(49\!\cdots\!42\)\( T^{4} + 89270113051354428 p^{18} T^{6} + p^{36} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 70065940 T + 75482606242699062 T^{2} + 70065940 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 169527552 T + 37765721149967198 T^{2} - 169527552 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 341453401608113868 T^{2} + \)\(66\!\cdots\!38\)\( T^{4} + 341453401608113868 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 617196131258526788 T^{2} + \)\(33\!\cdots\!22\)\( T^{4} + 617196131258526788 p^{18} T^{6} + p^{36} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 373066316 T + 1349771209210272294 T^{2} - 373066316 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
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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
−7.34752801424975605776144832353, −6.64400857409248575181857062468, −6.46369056415544917626056397011, −6.43673236588647085659940125801, −6.31886471773752504146191539834, −5.69622824368094164320353635489, −5.45988182738649975325955074911, −5.44857522228645728991137568154, −5.31500323305269580650273863902, −4.46286209474309887253657692816, −4.23858363512590424630182744719, −4.19575320910069221435245489294, −4.06557143107376200310592401691, −3.33870418915209804905270689672, −3.28330156913512780354451349021, −3.16009290763369516171001878644, −2.85974398931334454493030256991, −2.07817387579199303012563667313, −2.00238376965459207325981190593, −1.96855535052053053500366287075, −1.70021522450126866163032505404, −0.887497855719078454569012937446, −0.67728431355162196511183671330, −0.43511990618449806590062873330, −0.32263902497228376182065600056,
0.32263902497228376182065600056, 0.43511990618449806590062873330, 0.67728431355162196511183671330, 0.887497855719078454569012937446, 1.70021522450126866163032505404, 1.96855535052053053500366287075, 2.00238376965459207325981190593, 2.07817387579199303012563667313, 2.85974398931334454493030256991, 3.16009290763369516171001878644, 3.28330156913512780354451349021, 3.33870418915209804905270689672, 4.06557143107376200310592401691, 4.19575320910069221435245489294, 4.23858363512590424630182744719, 4.46286209474309887253657692816, 5.31500323305269580650273863902, 5.44857522228645728991137568154, 5.45988182738649975325955074911, 5.69622824368094164320353635489, 6.31886471773752504146191539834, 6.43673236588647085659940125801, 6.46369056415544917626056397011, 6.64400857409248575181857062468, 7.34752801424975605776144832353
