| L(s) = 1 | − 6·9-s + 56·17-s − 4·25-s + 56·41-s − 92·49-s + 200·73-s + 27·81-s − 248·89-s − 584·97-s + 520·113-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 336·153-s + 157-s + 163-s + 167-s − 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 3.29·17-s − 0.159·25-s + 1.36·41-s − 1.87·49-s + 2.73·73-s + 1/3·81-s − 2.78·89-s − 6.02·97-s + 4.60·113-s + 3.20·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 2.19·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.422324410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.422324410\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 3650 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 766 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1730 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 4610 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 4370 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 866 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 9506 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 13346 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 146 T + p^{2} T^{2} )^{4} \) |
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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
−7.24322191446746078333138259386, −7.23253419312110200561860400416, −6.63728602334032177810030237178, −6.48816480053014858610863140966, −6.37186570097152419273737187589, −5.83484671016590661978701547382, −5.69106777963148823299799601167, −5.59951040785179314968986217989, −5.45704107329987804436120806353, −5.28978005120934197552998741196, −4.61393816035973702860807141686, −4.60077143357596360096136212388, −4.46851950312496309022388011931, −3.88005144755581247212850530051, −3.76953961486365311975113178046, −3.25629167577844392518805992396, −3.24250310819799358439606180288, −3.05992712150639401627166108926, −2.66744169760783635816810187278, −2.27985798001979513732326838678, −1.81573815139331683759911593055, −1.61103083601623000909648284811, −0.994552364913843698085193449956, −0.895480285921050070660857141462, −0.32552671190579768496085932262,
0.32552671190579768496085932262, 0.895480285921050070660857141462, 0.994552364913843698085193449956, 1.61103083601623000909648284811, 1.81573815139331683759911593055, 2.27985798001979513732326838678, 2.66744169760783635816810187278, 3.05992712150639401627166108926, 3.24250310819799358439606180288, 3.25629167577844392518805992396, 3.76953961486365311975113178046, 3.88005144755581247212850530051, 4.46851950312496309022388011931, 4.60077143357596360096136212388, 4.61393816035973702860807141686, 5.28978005120934197552998741196, 5.45704107329987804436120806353, 5.59951040785179314968986217989, 5.69106777963148823299799601167, 5.83484671016590661978701547382, 6.37186570097152419273737187589, 6.48816480053014858610863140966, 6.63728602334032177810030237178, 7.23253419312110200561860400416, 7.24322191446746078333138259386
