| L(s) = 1 | + 6·9-s − 4·11-s + 12·19-s + 32·29-s − 8·31-s + 20·41-s + 4·49-s + 8·59-s − 24·61-s + 16·71-s + 40·79-s + 17·81-s − 36·89-s − 24·99-s + 8·101-s + 24·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.20·11-s + 2.75·19-s + 5.94·29-s − 1.43·31-s + 3.12·41-s + 4/7·49-s + 1.04·59-s − 3.07·61-s + 1.89·71-s + 4.50·79-s + 17/9·81-s − 3.81·89-s − 2.41·99-s + 0.796·101-s + 2.29·109-s + 2/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.19696189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.19696189\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 19 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 579 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 45 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10006 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 9939 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 258 T^{2} + 27011 T^{4} - 258 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 250 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 278 T^{2} + 32451 T^{4} - 278 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 18 T + 251 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1286 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.32595186564942816715541268597, −5.80153317839184162023763078696, −5.63087714976238310658672095417, −5.56132818329605838215511769342, −5.41334783062801431292329675295, −4.96697952935125233004195013578, −4.85755038359512127093859230172, −4.78945963628878229836538094404, −4.46274492180298973954083376472, −4.37796570950588149668113930921, −4.24047366633191275431804908420, −3.83807818552268926507835528631, −3.70228757176455576848050922442, −3.24587057717606327710988706230, −3.06834135097789421652680552866, −2.97135219300863248273948531392, −2.87260710518587221396619414197, −2.43362422814152774647991266828, −2.16891733927570111318208587569, −2.02698896702243112711522493667, −1.50994492667170263219602195539, −1.29142125061719668563354497894, −0.798829048006689492730289745819, −0.77955967243387585716311151967, −0.70762900335616452250646992932,
0.70762900335616452250646992932, 0.77955967243387585716311151967, 0.798829048006689492730289745819, 1.29142125061719668563354497894, 1.50994492667170263219602195539, 2.02698896702243112711522493667, 2.16891733927570111318208587569, 2.43362422814152774647991266828, 2.87260710518587221396619414197, 2.97135219300863248273948531392, 3.06834135097789421652680552866, 3.24587057717606327710988706230, 3.70228757176455576848050922442, 3.83807818552268926507835528631, 4.24047366633191275431804908420, 4.37796570950588149668113930921, 4.46274492180298973954083376472, 4.78945963628878229836538094404, 4.85755038359512127093859230172, 4.96697952935125233004195013578, 5.41334783062801431292329675295, 5.56132818329605838215511769342, 5.63087714976238310658672095417, 5.80153317839184162023763078696, 6.32595186564942816715541268597
