| L(s) = 1 | − 4·7-s − 4·9-s − 12·19-s + 10·25-s + 20·43-s − 18·49-s − 4·61-s + 16·63-s − 12·73-s + 7·81-s + 34·121-s + 127-s + 131-s + 48·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 48·171-s + 173-s − 40·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 4/3·9-s − 2.75·19-s + 2·25-s + 3.04·43-s − 2.57·49-s − 0.512·61-s + 2.01·63-s − 1.40·73-s + 7/9·81-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 4.16·133-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 + 2.46·169-s + 3.67·171-s + 0.0760·173-s − 3.02·175-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256473471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256473471\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 89 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 184 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.03478316735657042958477977067, −6.98132831895399362915872014492, −6.74268050916995412554314943872, −6.61383618787241119710815369609, −6.29168550720794500664621359888, −6.06644051692345301172136881689, −5.97421087106223824009225680366, −5.68984777965860849972952620676, −5.62277476298258352768183391930, −4.99759909603623267542343448495, −4.99525727939636799975678945713, −4.62411563386382750183457771526, −4.19514958909161710162985748078, −4.17101021529907217688289906249, −4.17062025127237771486620201882, −3.30853017385887234020676798615, −3.17379438390478724855635606705, −3.10501960608518363710646247425, −2.98842393226924914420396648086, −2.35457229626279657845647034775, −2.27218448789381555023435913855, −1.85105490044898194042503456436, −1.42286338387552004725334514200, −0.56269559371047578467566630843, −0.44714780020650638930014820179,
0.44714780020650638930014820179, 0.56269559371047578467566630843, 1.42286338387552004725334514200, 1.85105490044898194042503456436, 2.27218448789381555023435913855, 2.35457229626279657845647034775, 2.98842393226924914420396648086, 3.10501960608518363710646247425, 3.17379438390478724855635606705, 3.30853017385887234020676798615, 4.17062025127237771486620201882, 4.17101021529907217688289906249, 4.19514958909161710162985748078, 4.62411563386382750183457771526, 4.99525727939636799975678945713, 4.99759909603623267542343448495, 5.62277476298258352768183391930, 5.68984777965860849972952620676, 5.97421087106223824009225680366, 6.06644051692345301172136881689, 6.29168550720794500664621359888, 6.61383618787241119710815369609, 6.74268050916995412554314943872, 6.98132831895399362915872014492, 7.03478316735657042958477977067
