| L(s) = 1 | − 2·3-s − 5·9-s − 20·13-s − 4·19-s + 18·25-s + 28·27-s + 44·31-s − 12·37-s + 40·39-s + 164·43-s + 8·57-s + 172·61-s + 4·67-s − 164·73-s − 36·75-s + 20·79-s − 11·81-s − 88·93-s + 188·97-s + 268·103-s + 20·109-s + 24·111-s + 100·117-s + 210·121-s + 127-s − 328·129-s + 131-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 5/9·9-s − 1.53·13-s − 0.210·19-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 + 8/57·57-s + 2.81·61-s + 4/67·67-s − 2.24·73-s − 0.479·75-s + 0.253·79-s − 0.135·81-s − 0.946·93-s + 1.93·97-s + 2.60·103-s + 0.183·109-s + 8/37·111-s + 0.854·117-s + 1.73·121-s + 0.00787·127-s − 2.54·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.821338949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821338949\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 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
−9.705742369557791484116920636740, −9.575697881950485442702339700806, −8.772914202252505087665193009923, −8.723734930495159636561748606810, −8.223967026506504283713956496418, −7.55221161419253050400809773489, −7.30453261614245276068656867358, −6.98244319821467629846908490729, −6.34641495465019141315356431912, −5.93464170940541968101130444680, −5.66356060041287039042793732552, −5.08426713406861487908877024997, −4.52325161320778741248828865678, −4.48790621766800915413295055725, −3.60454640769073190694486672655, −2.96863840142697748307446282849, −2.48269222113906623690332212663, −2.10847724879635944579758693494, −0.874890692367337895974216340922, −0.56130893396532418577808620310,
0.56130893396532418577808620310, 0.874890692367337895974216340922, 2.10847724879635944579758693494, 2.48269222113906623690332212663, 2.96863840142697748307446282849, 3.60454640769073190694486672655, 4.48790621766800915413295055725, 4.52325161320778741248828865678, 5.08426713406861487908877024997, 5.66356060041287039042793732552, 5.93464170940541968101130444680, 6.34641495465019141315356431912, 6.98244319821467629846908490729, 7.30453261614245276068656867358, 7.55221161419253050400809773489, 8.223967026506504283713956496418, 8.723734930495159636561748606810, 8.772914202252505087665193009923, 9.575697881950485442702339700806, 9.705742369557791484116920636740
