| L(s) = 1 | − 2·3-s + 2·7-s − 5·9-s + 30·13-s − 46·19-s − 4·21-s + 28·27-s + 66·31-s + 132·37-s − 60·39-s − 14·43-s − 95·49-s + 92·57-s + 78·61-s − 10·63-s + 226·67-s + 116·73-s + 140·79-s − 11·81-s + 60·91-s − 132·93-s − 2·97-s − 52·103-s − 290·109-s − 264·111-s − 150·117-s + 170·121-s + ⋯ |
| L(s) = 1 | − 2/3·3-s + 2/7·7-s − 5/9·9-s + 2.30·13-s − 2.42·19-s − 0.190·21-s + 1.03·27-s + 2.12·31-s + 3.56·37-s − 1.53·39-s − 0.325·43-s − 1.93·49-s + 1.61·57-s + 1.27·61-s − 0.158·63-s + 3.37·67-s + 1.58·73-s + 1.77·79-s − 0.135·81-s + 0.659·91-s − 1.41·93-s − 0.0206·97-s − 0.504·103-s − 2.66·109-s − 2.37·111-s − 1.28·117-s + 1.40·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.034623088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034623088\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 2 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 170 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 15 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 186 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1050 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1034 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 33 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 66 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2010 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2370 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4266 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3406 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 113 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9434 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 7650 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
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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
−10.78603747719555438479941489862, −10.48927895489790962641216564375, −9.665764654666139366817334466355, −9.580494958637741416985750891609, −8.746868041827164753270475515836, −8.366551235310985887091757988732, −8.052280565635640672839873265614, −8.020588136784726195171504760540, −6.66635371291722039293820751044, −6.49294746680711043076987404803, −6.28023929042224367139618703498, −5.87362023835728636415512763110, −5.06129784912475892808777745490, −4.72727306270985088247752373384, −3.86515372339682886536851062679, −3.86336414197754031827621016960, −2.70410938513859598158611091180, −2.30981246301369558959159593755, −1.20986466897481713854377244665, −0.64350695186992596231213702013,
0.64350695186992596231213702013, 1.20986466897481713854377244665, 2.30981246301369558959159593755, 2.70410938513859598158611091180, 3.86336414197754031827621016960, 3.86515372339682886536851062679, 4.72727306270985088247752373384, 5.06129784912475892808777745490, 5.87362023835728636415512763110, 6.28023929042224367139618703498, 6.49294746680711043076987404803, 6.66635371291722039293820751044, 8.020588136784726195171504760540, 8.052280565635640672839873265614, 8.366551235310985887091757988732, 8.746868041827164753270475515836, 9.580494958637741416985750891609, 9.665764654666139366817334466355, 10.48927895489790962641216564375, 10.78603747719555438479941489862
