| L(s) = 1 | + 10·4-s + 43·16-s + 22·25-s + 32·37-s − 16·43-s + 20·64-s + 416·67-s + 44·79-s + 220·100-s − 544·109-s − 458·121-s + 127-s + 131-s + 137-s + 139-s + 320·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 460·169-s − 160·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 2.68·16-s + 0.879·25-s + 0.864·37-s − 0.372·43-s + 5/16·64-s + 6.20·67-s + 0.556·79-s + 11/5·100-s − 4.99·109-s − 3.78·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.16·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.72·169-s − 0.930·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(7.981189536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.981189536\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 11 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 229 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 1006 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1045 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1055 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1958 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 4262 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1259 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 371 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 5990 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 1294 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 8306 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 3421 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 3034 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17495 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.83107830490850512224951325027, −7.74862301679821688847674139461, −7.17374314870177987214517735467, −6.81167392232529196459848282834, −6.79520819380049887146932810162, −6.63451545672068266481773517940, −6.62487972068021594723531341346, −6.32555600423583309055954377097, −5.77480062922273107269941084896, −5.57224531861615805863368128323, −5.37682481761931528355768967985, −5.15739016861448300759667820788, −4.88611763785054736502510221530, −4.16458182831732667743344377756, −4.08638030768795206436123964925, −4.05408299348812360341818975251, −3.17518143686522664558468157001, −3.14486029641606332509999239047, −2.97757186779295188046402655450, −2.33366727015342681627704287155, −2.17223857071201286041439031901, −2.10336457900320394978157455779, −1.41766070350080452261424857013, −1.06557036327455814855661028119, −0.50299448767447714565171089025,
0.50299448767447714565171089025, 1.06557036327455814855661028119, 1.41766070350080452261424857013, 2.10336457900320394978157455779, 2.17223857071201286041439031901, 2.33366727015342681627704287155, 2.97757186779295188046402655450, 3.14486029641606332509999239047, 3.17518143686522664558468157001, 4.05408299348812360341818975251, 4.08638030768795206436123964925, 4.16458182831732667743344377756, 4.88611763785054736502510221530, 5.15739016861448300759667820788, 5.37682481761931528355768967985, 5.57224531861615805863368128323, 5.77480062922273107269941084896, 6.32555600423583309055954377097, 6.62487972068021594723531341346, 6.63451545672068266481773517940, 6.79520819380049887146932810162, 6.81167392232529196459848282834, 7.17374314870177987214517735467, 7.74862301679821688847674139461, 7.83107830490850512224951325027
