L(s) = 1 | + 4·4-s + 12·16-s + 8·31-s − 28·49-s + 32·64-s − 56·79-s + 4·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 112·196-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 2·4-s + 3·16-s + 1.43·31-s − 4·49-s + 4·64-s − 6.30·79-s + 4/11·121-s + 2.87·124-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 − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 8·196-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8887323399\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8887323399\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 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
−6.79136744912557205867992485068, −6.41224646680758118439370590761, −5.99192804942718422061768364657, −5.95797769554445600635098437557, −5.92232556484349220390557285734, −5.90842224045289553594375133979, −5.26724058838719926595848995352, −5.09785867125132723204938054807, −4.94491453448288443906406302958, −4.66309656866423096409596836507, −4.47753294219461311597704097934, −4.22620944874835906214169105691, −3.85517197682056741611312372326, −3.49739731886081680667106932138, −3.39395748091631167140877406231, −3.29376860517439215172738799966, −2.82831439352488649218422623630, −2.60570161908968463137462635355, −2.50943519037463279362645708739, −2.27211372554076332137311828505, −1.78207355844286769919428610011, −1.36840164241791941006126396385, −1.25508048454219950982752110544, −1.24025684309267722243947601469, −0.13086096771456896026793793062,
0.13086096771456896026793793062, 1.24025684309267722243947601469, 1.25508048454219950982752110544, 1.36840164241791941006126396385, 1.78207355844286769919428610011, 2.27211372554076332137311828505, 2.50943519037463279362645708739, 2.60570161908968463137462635355, 2.82831439352488649218422623630, 3.29376860517439215172738799966, 3.39395748091631167140877406231, 3.49739731886081680667106932138, 3.85517197682056741611312372326, 4.22620944874835906214169105691, 4.47753294219461311597704097934, 4.66309656866423096409596836507, 4.94491453448288443906406302958, 5.09785867125132723204938054807, 5.26724058838719926595848995352, 5.90842224045289553594375133979, 5.92232556484349220390557285734, 5.95797769554445600635098437557, 5.99192804942718422061768364657, 6.41224646680758118439370590761, 6.79136744912557205867992485068