| L(s) = 1 | + 2·3-s − 12·7-s − 5·9-s + 20·13-s + 4·19-s − 24·21-s + 18·25-s − 28·27-s − 44·31-s − 12·37-s + 40·39-s + 164·43-s + 10·49-s + 8·57-s − 172·61-s + 60·63-s + 4·67-s + 164·73-s + 36·75-s + 20·79-s − 11·81-s − 240·91-s − 88·93-s − 188·97-s − 268·103-s + 20·109-s − 24·111-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 1.71·7-s − 5/9·9-s + 1.53·13-s + 4/19·19-s − 8/7·21-s + 0.719·25-s − 1.03·27-s − 1.41·31-s − 0.324·37-s + 1.02·39-s + 3.81·43-s + 0.204·49-s + 8/57·57-s − 2.81·61-s + 0.952·63-s + 4/67·67-s + 2.24·73-s + 0.479·75-s + 0.253·79-s − 0.135·81-s − 2.63·91-s − 0.946·93-s − 1.93·97-s − 2.60·103-s + 0.183·109-s − 0.216·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8882750702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8882750702\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 66 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 930 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1394 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1746 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1554 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 86 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 5406 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8370 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14690 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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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
−17.93329962250395857674990055759, −17.14668355749950927793260670781, −16.28425281770056576603944690490, −16.16509422432547218507615041236, −15.41817941221427843923117773211, −14.73168390092593549178382905593, −13.77100009717908332975017242392, −13.70006705163429983926489973324, −12.61782911242619031988447152576, −12.46828269382948081457220826153, −10.99342490422542368991926475278, −10.84205192665977700251879871973, −9.396373175556693518577016958289, −9.311841927145405240809802494724, −8.396629698352459685727855168211, −7.40713148646346123493409827011, −6.36573606719747848689709829916, −5.72798389202971793650888671793, −3.84755496424067939340094305126, −2.98246483718788002904300333538,
2.98246483718788002904300333538, 3.84755496424067939340094305126, 5.72798389202971793650888671793, 6.36573606719747848689709829916, 7.40713148646346123493409827011, 8.396629698352459685727855168211, 9.311841927145405240809802494724, 9.396373175556693518577016958289, 10.84205192665977700251879871973, 10.99342490422542368991926475278, 12.46828269382948081457220826153, 12.61782911242619031988447152576, 13.70006705163429983926489973324, 13.77100009717908332975017242392, 14.73168390092593549178382905593, 15.41817941221427843923117773211, 16.16509422432547218507615041236, 16.28425281770056576603944690490, 17.14668355749950927793260670781, 17.93329962250395857674990055759
