| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.935212402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.935212402\) |
| \(L(2)\) |
|
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
−9.699957338569876494983695929144, −9.334285197994046275524173331425, −9.033593755516038446616078043060, −8.387389895741237391503201003640, −8.341304345896403867904511079030, −7.57448149732206533313881485549, −7.57208059946903596252165347780, −6.98275791236992341517807453868, −6.16010612431649680584595721204, −6.06582711817808962395285935207, −5.68051544359837898186335518537, −5.19592167585972882283964559445, −4.50293936565785692768583496580, −3.86809375268879681009125346010, −3.61037811595556285241351476240, −2.92535374726308353740775068522, −2.90004332520447258714934153734, −1.79843764510501858463619583484, −1.29030340834811466273964944127, −0.60949043483273777083304318518,
0.60949043483273777083304318518, 1.29030340834811466273964944127, 1.79843764510501858463619583484, 2.90004332520447258714934153734, 2.92535374726308353740775068522, 3.61037811595556285241351476240, 3.86809375268879681009125346010, 4.50293936565785692768583496580, 5.19592167585972882283964559445, 5.68051544359837898186335518537, 6.06582711817808962395285935207, 6.16010612431649680584595721204, 6.98275791236992341517807453868, 7.57208059946903596252165347780, 7.57448149732206533313881485549, 8.341304345896403867904511079030, 8.387389895741237391503201003640, 9.033593755516038446616078043060, 9.334285197994046275524173331425, 9.699957338569876494983695929144
