L(s) = 1 | + 8·13-s + 4·17-s + 4·19-s + 4·23-s + 5·25-s − 4·29-s − 8·31-s + 6·37-s − 24·41-s + 8·43-s − 8·47-s + 6·53-s + 12·59-s − 4·61-s + 4·67-s + 24·71-s − 8·73-s + 16·79-s + 8·83-s + 4·89-s + 32·97-s − 8·101-s + 8·103-s + 8·107-s + 14·109-s − 4·113-s + 11·121-s + ⋯ |
L(s) = 1 | + 2.21·13-s + 0.970·17-s + 0.917·19-s + 0.834·23-s + 25-s − 0.742·29-s − 1.43·31-s + 0.986·37-s − 3.74·41-s + 1.21·43-s − 1.16·47-s + 0.824·53-s + 1.56·59-s − 0.512·61-s + 0.488·67-s + 2.84·71-s − 0.936·73-s + 1.80·79-s + 0.878·83-s + 0.423·89-s + 3.24·97-s − 0.796·101-s + 0.788·103-s + 0.773·107-s + 1.34·109-s − 0.376·113-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.115681547\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.115681547\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−8.702588209118105632258166055804, −8.463737818205991490065134747749, −8.003974949280172331796878074685, −7.76400294670509964441091565995, −7.15527299740182421070380344006, −6.98800646494359594423121897168, −6.48330950749577220097973185935, −6.21766149567369653823178675441, −5.66342594951198958949434724278, −5.42190027010768857994587264283, −4.98858734050803889955053387291, −4.72974496170981752900770649996, −3.82163768734885911096073538304, −3.67376064815475273490804968992, −3.27444153123625783695016518596, −3.10725749204855986555159595390, −1.94089756562227612508501279134, −1.93278269196804540520738321602, −0.907692272005537353040256708182, −0.836292010175742533646340816432,
0.836292010175742533646340816432, 0.907692272005537353040256708182, 1.93278269196804540520738321602, 1.94089756562227612508501279134, 3.10725749204855986555159595390, 3.27444153123625783695016518596, 3.67376064815475273490804968992, 3.82163768734885911096073538304, 4.72974496170981752900770649996, 4.98858734050803889955053387291, 5.42190027010768857994587264283, 5.66342594951198958949434724278, 6.21766149567369653823178675441, 6.48330950749577220097973185935, 6.98800646494359594423121897168, 7.15527299740182421070380344006, 7.76400294670509964441091565995, 8.003974949280172331796878074685, 8.463737818205991490065134747749, 8.702588209118105632258166055804