| L(s) = 1 | − 2·5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s + 4·89-s + 4·101-s + 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·5-s + 25-s − 2·29-s + 2·41-s − 49-s − 2·61-s + 4·89-s + 4·101-s + 4·109-s − 2·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6674095859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6674095859\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.70061787975834571015686303652, −6.36389806479051389419857020897, −6.12640977067959197527110102804, −6.10152030812417905127427122256, −5.80882538162803068541835881723, −5.73698672350256743905285147228, −5.48617037245777466407456636448, −4.94316220882254008543057175965, −4.76348182084597636064235446332, −4.72765627252375934363096693239, −4.66258351014817769349426701055, −4.35368246118398338684239370523, −3.95832788026268393827972234958, −3.68691545613280419740794780920, −3.57165095179946575183583520492, −3.47721463976858957705235265307, −3.45474718625286887030640649981, −2.81541551353008979559568852167, −2.66039398739313116975375322202, −2.38439582429377656778591876945, −1.88191527200502079658863358174, −1.85507581649513027495424109356, −1.48127986216760581996466882832, −0.76524831502332005118360907554, −0.55141582021464210397343619677,
0.55141582021464210397343619677, 0.76524831502332005118360907554, 1.48127986216760581996466882832, 1.85507581649513027495424109356, 1.88191527200502079658863358174, 2.38439582429377656778591876945, 2.66039398739313116975375322202, 2.81541551353008979559568852167, 3.45474718625286887030640649981, 3.47721463976858957705235265307, 3.57165095179946575183583520492, 3.68691545613280419740794780920, 3.95832788026268393827972234958, 4.35368246118398338684239370523, 4.66258351014817769349426701055, 4.72765627252375934363096693239, 4.76348182084597636064235446332, 4.94316220882254008543057175965, 5.48617037245777466407456636448, 5.73698672350256743905285147228, 5.80882538162803068541835881723, 6.10152030812417905127427122256, 6.12640977067959197527110102804, 6.36389806479051389419857020897, 6.70061787975834571015686303652
