L(s) = 1 | − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | − 4·9-s + 2·11-s − 2·19-s − 2·41-s + 49-s − 2·59-s + 10·81-s + 2·89-s − 8·99-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 8·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1210823462\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1210823462\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 59 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.23723442122574676473765063285, −5.83882136065378387837589732836, −5.75608960533784646845657061716, −5.65174714330096384006790224702, −5.57721357462453009047090832336, −5.15339221181815923094476856809, −5.08971323717827158839558591371, −4.62740817848954176014717518225, −4.62108847752277971524425784973, −4.48465535260203657476214622750, −4.16410299623851445200865800315, −3.79257508186621277055952131660, −3.66849357286705439055417409692, −3.39993831743869432911254259836, −3.27945918539258542339187992909, −3.25454378175731826934697389640, −2.76888523229911173567052149398, −2.63438398402300114061068444108, −2.27878941582784016870294192260, −2.16660993313762805158250279725, −1.97396195465770005998945690585, −1.63929935785592869824700844551, −1.04841541239011216516111618840, −1.00443465304627483002935169179, −0.13130051037305364865967451466,
0.13130051037305364865967451466, 1.00443465304627483002935169179, 1.04841541239011216516111618840, 1.63929935785592869824700844551, 1.97396195465770005998945690585, 2.16660993313762805158250279725, 2.27878941582784016870294192260, 2.63438398402300114061068444108, 2.76888523229911173567052149398, 3.25454378175731826934697389640, 3.27945918539258542339187992909, 3.39993831743869432911254259836, 3.66849357286705439055417409692, 3.79257508186621277055952131660, 4.16410299623851445200865800315, 4.48465535260203657476214622750, 4.62108847752277971524425784973, 4.62740817848954176014717518225, 5.08971323717827158839558591371, 5.15339221181815923094476856809, 5.57721357462453009047090832336, 5.65174714330096384006790224702, 5.75608960533784646845657061716, 5.83882136065378387837589732836, 6.23723442122574676473765063285