| L(s) = 1 | − 4·7-s − 2·9-s + 8·17-s + 4·23-s − 6·25-s + 8·31-s + 16·41-s + 10·49-s + 8·63-s − 32·71-s + 24·73-s + 2·81-s + 32·89-s + 16·97-s − 24·103-s + 36·113-s − 32·119-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s − 16·161-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 2/3·9-s + 1.94·17-s + 0.834·23-s − 6/5·25-s + 1.43·31-s + 2.49·41-s + 10/7·49-s + 1.00·63-s − 3.79·71-s + 2.80·73-s + 2/9·81-s + 3.39·89-s + 1.62·97-s − 2.36·103-s + 3.38·113-s − 2.93·119-s − 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.29·153-s + 0.0798·157-s − 1.26·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.411081386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.411081386\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 6 T^{2} + 42 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 38 T^{2} + 682 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 66 T^{2} + 1794 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 76 T^{2} + 2854 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 108 T^{2} + 5382 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 152 T^{2} + 9406 T^{4} + 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 132 T^{2} + 8886 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 178 T^{2} + 14050 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 230 T^{2} + 20650 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 112 T^{2} + 8782 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 210 T^{2} + 23970 T^{4} + 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.57151627396023210922579479916, −6.34808584481768196951168943530, −6.07008632277503567832840651619, −5.99602798077333743772344328863, −5.91991487950114033278448733696, −5.57390904694470638065154291277, −5.35049596646339340638296691584, −5.25172061616993512094772609216, −4.85841454062855761493651590840, −4.68404508156184007866714252843, −4.33911346829486591503435058394, −4.12768848186133059959467983526, −4.07144676215838002184252540556, −3.47316049738836127794905085806, −3.37586428681008076109563540564, −3.24949886372362640930594598836, −3.17920930014929574114611450311, −2.65364862048000860938992899901, −2.47521103627974456287929866930, −2.25839757387123034607915633851, −1.97791570942708712065464181645, −1.25223851808772694937422300766, −1.17598463899746830904733288293, −0.72822741090773578514497926136, −0.36139886898797756619735050082,
0.36139886898797756619735050082, 0.72822741090773578514497926136, 1.17598463899746830904733288293, 1.25223851808772694937422300766, 1.97791570942708712065464181645, 2.25839757387123034607915633851, 2.47521103627974456287929866930, 2.65364862048000860938992899901, 3.17920930014929574114611450311, 3.24949886372362640930594598836, 3.37586428681008076109563540564, 3.47316049738836127794905085806, 4.07144676215838002184252540556, 4.12768848186133059959467983526, 4.33911346829486591503435058394, 4.68404508156184007866714252843, 4.85841454062855761493651590840, 5.25172061616993512094772609216, 5.35049596646339340638296691584, 5.57390904694470638065154291277, 5.91991487950114033278448733696, 5.99602798077333743772344328863, 6.07008632277503567832840651619, 6.34808584481768196951168943530, 6.57151627396023210922579479916
