| L(s) = 1 | − 16·4-s + 192·16-s + 1.57e3·25-s − 2.40e3·37-s − 4.64e3·43-s − 2.04e3·64-s + 3.31e4·67-s + 1.23e4·79-s − 2.51e4·100-s + 6.83e4·109-s + 5.18e4·121-s + 127-s + 131-s + 137-s + 139-s + 3.84e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.13e4·169-s + 7.42e4·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 3/4·16-s + 2.51·25-s − 1.75·37-s − 2.50·43-s − 1/2·64-s + 7.37·67-s + 1.97·79-s − 2.51·100-s + 5.75·109-s + 3.54·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 1.75·148-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2.49·169-s + 2.50·172-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(9.037237094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.037237094\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 786 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 25920 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 35678 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 62642 T^{2} + p^{8} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 237442 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 269360 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1407600 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 1267042 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 600 T + p^{4} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3041522 T^{2} + p^{8} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 1160 T + p^{4} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 8599362 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 14561040 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10825122 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 20986882 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8278 T + p^{4} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 47438160 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 29530718 T^{2} + p^{8} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3082 T + p^{4} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 89244958 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 121296882 T^{2} + p^{8} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 176478562 T^{2} + p^{8} T^{4} )^{2} \) |
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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
−6.84186939513581472710495659672, −6.50304519281623887603441037980, −6.20148479236185417276057188814, −6.03515767951542469027152161018, −5.68215451012021899402700202690, −5.37406923649013552214323095183, −5.15019877649173145700739231750, −4.99671166967939936994122519644, −4.98639816092100076646509884132, −4.62906824658652018147000950859, −4.36416300899380324020028220677, −4.07192147873446100299422192472, −3.64799280207427927223877316871, −3.60742442495824553072968936075, −3.18284466812979469483569494879, −3.15867888467095128416955548410, −2.94244776727246262523199655728, −2.30067314370550284374548266362, −1.90496888449614744469087518679, −1.82295189304410097979574740647, −1.80801812595178563220882521624, −0.74760257563252851908536910072, −0.71708373298308893948165357423, −0.71554486659648305745554866947, −0.46086984751355748368224355813,
0.46086984751355748368224355813, 0.71554486659648305745554866947, 0.71708373298308893948165357423, 0.74760257563252851908536910072, 1.80801812595178563220882521624, 1.82295189304410097979574740647, 1.90496888449614744469087518679, 2.30067314370550284374548266362, 2.94244776727246262523199655728, 3.15867888467095128416955548410, 3.18284466812979469483569494879, 3.60742442495824553072968936075, 3.64799280207427927223877316871, 4.07192147873446100299422192472, 4.36416300899380324020028220677, 4.62906824658652018147000950859, 4.98639816092100076646509884132, 4.99671166967939936994122519644, 5.15019877649173145700739231750, 5.37406923649013552214323095183, 5.68215451012021899402700202690, 6.03515767951542469027152161018, 6.20148479236185417276057188814, 6.50304519281623887603441037980, 6.84186939513581472710495659672
