| L(s) = 1 | + 4-s − 4·7-s − 2·9-s − 11-s + 16-s + 2·25-s − 4·28-s + 12·29-s − 2·36-s − 8·37-s − 8·43-s − 44-s + 9·49-s + 8·63-s + 64-s + 4·67-s + 12·71-s + 4·77-s − 8·79-s − 5·81-s + 2·99-s + 2·100-s − 12·107-s + 16·109-s − 4·112-s + 12·113-s + 12·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.51·7-s − 2/3·9-s − 0.301·11-s + 1/4·16-s + 2/5·25-s − 0.755·28-s + 2.22·29-s − 1/3·36-s − 1.31·37-s − 1.21·43-s − 0.150·44-s + 9/7·49-s + 1.00·63-s + 1/8·64-s + 0.488·67-s + 1.42·71-s + 0.455·77-s − 0.900·79-s − 5/9·81-s + 0.201·99-s + 1/5·100-s − 1.16·107-s + 1.53·109-s − 0.377·112-s + 1.12·113-s + 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6353160322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6353160322\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−13.14444196011017736624954771284, −12.58572238817061641688555604504, −12.07640046661767485447135762696, −11.48746584495738347692356080454, −10.61647521625764182154291152125, −10.18774784796246493803027109873, −9.571837480671157198635959777949, −8.691923532445567953257811423624, −8.199522970356618712223090269825, −7.04378127151225446544122784756, −6.61188575429027541858457662119, −5.88669620904765581852775751295, −4.91759000072777764587960332982, −3.49031776546888561563567080235, −2.71835047120874446997595797974,
2.71835047120874446997595797974, 3.49031776546888561563567080235, 4.91759000072777764587960332982, 5.88669620904765581852775751295, 6.61188575429027541858457662119, 7.04378127151225446544122784756, 8.199522970356618712223090269825, 8.691923532445567953257811423624, 9.571837480671157198635959777949, 10.18774784796246493803027109873, 10.61647521625764182154291152125, 11.48746584495738347692356080454, 12.07640046661767485447135762696, 12.58572238817061641688555604504, 13.14444196011017736624954771284
