| L(s) = 1 | + 2·3-s + 9-s − 6·11-s + 3·13-s + 2·25-s − 4·27-s − 12·33-s − 8·37-s + 6·39-s − 6·47-s + 2·49-s + 18·59-s − 20·61-s + 6·71-s + 16·73-s + 4·75-s − 11·81-s + 6·83-s + 4·97-s − 6·99-s + 12·107-s + 4·109-s − 16·111-s + 3·117-s + 14·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.80·11-s + 0.832·13-s + 2/5·25-s − 0.769·27-s − 2.08·33-s − 1.31·37-s + 0.960·39-s − 0.875·47-s + 2/7·49-s + 2.34·59-s − 2.56·61-s + 0.712·71-s + 1.87·73-s + 0.461·75-s − 1.22·81-s + 0.658·83-s + 0.406·97-s − 0.603·99-s + 1.16·107-s + 0.383·109-s − 1.51·111-s + 0.277·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198190650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198190650\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.83938386150278550919213573993, −11.10396667697402235230386137033, −10.63320111520649592426179360099, −10.12772605802792931846181286551, −9.434192993168675227456234520967, −8.787041742505869349065801840331, −8.314896165317097370956884144463, −7.86686270169212177964764288118, −7.24296913197291863639486403305, −6.38405899423022690876523121871, −5.50424102239527159106193376074, −4.89774820299342356065803560786, −3.73226663070959059888191968994, −3.04830443487435851075515691327, −2.13934100360356323241190826970,
2.13934100360356323241190826970, 3.04830443487435851075515691327, 3.73226663070959059888191968994, 4.89774820299342356065803560786, 5.50424102239527159106193376074, 6.38405899423022690876523121871, 7.24296913197291863639486403305, 7.86686270169212177964764288118, 8.314896165317097370956884144463, 8.787041742505869349065801840331, 9.434192993168675227456234520967, 10.12772605802792931846181286551, 10.63320111520649592426179360099, 11.10396667697402235230386137033, 11.83938386150278550919213573993
