| L(s) = 1 | + 54·3-s − 3.44e4·7-s − 5.61e4·9-s − 3.39e5·13-s + 1.89e6·19-s − 1.86e6·21-s + 1.14e7·25-s − 6.21e6·27-s + 5.95e7·31-s − 1.21e8·37-s − 1.83e7·39-s − 2.14e8·43-s + 3.26e8·49-s + 1.02e8·57-s + 2.06e9·61-s + 1.93e9·63-s − 3.75e9·67-s − 5.69e9·73-s + 6.17e8·75-s − 2.97e9·79-s + 2.97e9·81-s + 1.16e10·91-s + 3.21e9·93-s − 3.18e9·97-s − 8.83e9·103-s − 3.09e10·109-s − 6.56e9·111-s + ⋯ |
| L(s) = 1 | + 2/9·3-s − 2.05·7-s − 0.950·9-s − 0.913·13-s + 0.766·19-s − 0.455·21-s + 1.17·25-s − 0.433·27-s + 2.08·31-s − 1.75·37-s − 0.203·39-s − 1.46·43-s + 1.15·49-s + 0.170·57-s + 2.44·61-s + 1.94·63-s − 2.78·67-s − 2.74·73-s + 0.260·75-s − 0.967·79-s + 0.854·81-s + 1.87·91-s + 0.462·93-s − 0.370·97-s − 0.762·103-s − 2.01·109-s − 0.389·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.09475208124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09475208124\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 - 2 p^{3} T + p^{10} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2288266 p T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2462 p T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 16976849522 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 169654 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3914170410818 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 949462 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 75790028393378 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 831288000078482 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 29793118 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 60811846 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6157996032931678 T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 107419706 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 33376598707262018 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 312876100791567218 T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 600827685707033522 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 1030793642 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1876742474 T + p^{10} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 690030731290713118 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2846528494 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1488647618 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 29429698146400299218 T^{2} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 26116812713945754722 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1592948926 T + p^{10} T^{2} )^{2} \) |
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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
−13.47537903319059814121962187680, −13.39610417721463844642477940939, −12.63549578279138392961755416051, −11.90715354257993683589529780157, −11.77927700823048858184059657163, −10.63028017744560555334890331561, −9.954505932868933990883062721774, −9.809300691048694345249303759104, −8.864939886294792979251355396805, −8.533713283089683671970236353929, −7.49941349000463127272866271553, −6.78761105883083281037208663999, −6.38166750940333190824141435422, −5.52176114248842933933409111705, −4.80538945293362044736615122124, −3.66561838892430051570399527320, −2.82969157455282525708069688712, −2.82100038869888219524701395950, −1.28940417259140117523932145212, −0.097515626278833738841075909351,
0.097515626278833738841075909351, 1.28940417259140117523932145212, 2.82100038869888219524701395950, 2.82969157455282525708069688712, 3.66561838892430051570399527320, 4.80538945293362044736615122124, 5.52176114248842933933409111705, 6.38166750940333190824141435422, 6.78761105883083281037208663999, 7.49941349000463127272866271553, 8.533713283089683671970236353929, 8.864939886294792979251355396805, 9.809300691048694345249303759104, 9.954505932868933990883062721774, 10.63028017744560555334890331561, 11.77927700823048858184059657163, 11.90715354257993683589529780157, 12.63549578279138392961755416051, 13.39610417721463844642477940939, 13.47537903319059814121962187680
