| L(s) = 1 | − 12·9-s + 10·25-s − 8·41-s + 12·49-s + 90·81-s + 56·89-s − 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 120·225-s + ⋯ |
| L(s) = 1 | − 4·9-s + 2·25-s − 1.24·41-s + 12/7·49-s + 10·81-s + 5.93·89-s − 3.27·121-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 + 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 − 8·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356053566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356053566\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69215013626618733383044319880, −7.48068970978746736046319225016, −7.18756683473603371820022916820, −6.73821421789768302157447690143, −6.56447127430724431282716170581, −6.45824831159538789471893326072, −6.17275619222713723282878027554, −5.92594989778316394919276623857, −5.74391522237832299648582130094, −5.38832197140397301019522503100, −5.20501338902674427373497473276, −5.05332110774645029716548460962, −4.86821913666920457197057313589, −4.52205616179358848688405552967, −4.05811579488334999495447688792, −3.50964589217780605333013792647, −3.45068251819807070971461564270, −3.43758080465563911755674165642, −2.77121120543910301898558455563, −2.61877163310922399718376574294, −2.54521300674090170171251205844, −2.09258949983626705239722401272, −1.54201255175378457978122028951, −0.71700709487751678555268787915, −0.48010892409400119125957988672,
0.48010892409400119125957988672, 0.71700709487751678555268787915, 1.54201255175378457978122028951, 2.09258949983626705239722401272, 2.54521300674090170171251205844, 2.61877163310922399718376574294, 2.77121120543910301898558455563, 3.43758080465563911755674165642, 3.45068251819807070971461564270, 3.50964589217780605333013792647, 4.05811579488334999495447688792, 4.52205616179358848688405552967, 4.86821913666920457197057313589, 5.05332110774645029716548460962, 5.20501338902674427373497473276, 5.38832197140397301019522503100, 5.74391522237832299648582130094, 5.92594989778316394919276623857, 6.17275619222713723282878027554, 6.45824831159538789471893326072, 6.56447127430724431282716170581, 6.73821421789768302157447690143, 7.18756683473603371820022916820, 7.48068970978746736046319225016, 7.69215013626618733383044319880
