L(s) = 1 | − 8·9-s − 48·41-s + 8·49-s + 30·81-s + 24·89-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 8/3·9-s − 7.49·41-s + 8/7·49-s + 10/3·81-s + 2.54·89-s + 4·121-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-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
\(L(1)\) |
\(\approx\) |
\(1.166224568\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.166224568\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.18095629099796538134315358796, −5.97811752126951766453453518295, −5.59111121396575116879731422709, −5.52956896075894788336641543047, −5.36549483512387828763251084359, −5.12648771918675255881103414058, −5.11023260360190335735175600501, −4.67344758068574341502654464187, −4.60297674271971492224693455695, −4.48218401815191520097636881223, −3.89686345521894590362533139084, −3.73833360125606421002503620570, −3.55252983173345136784038285130, −3.30562233966033094062972628984, −3.27172293108788720407437899697, −2.94623922898960700624088148944, −2.82051180885537447940440090741, −2.52483189858772914923008903234, −2.03394137902881336513794194195, −1.98181800414626140066174389166, −1.71702779871546634814321851182, −1.53881792470247520344419201980, −0.860988313887004135081495737893, −0.47868859532977740265935942113, −0.26115774685772823225426352013,
0.26115774685772823225426352013, 0.47868859532977740265935942113, 0.860988313887004135081495737893, 1.53881792470247520344419201980, 1.71702779871546634814321851182, 1.98181800414626140066174389166, 2.03394137902881336513794194195, 2.52483189858772914923008903234, 2.82051180885537447940440090741, 2.94623922898960700624088148944, 3.27172293108788720407437899697, 3.30562233966033094062972628984, 3.55252983173345136784038285130, 3.73833360125606421002503620570, 3.89686345521894590362533139084, 4.48218401815191520097636881223, 4.60297674271971492224693455695, 4.67344758068574341502654464187, 5.11023260360190335735175600501, 5.12648771918675255881103414058, 5.36549483512387828763251084359, 5.52956896075894788336641543047, 5.59111121396575116879731422709, 5.97811752126951766453453518295, 6.18095629099796538134315358796