| L(s) = 1 | − 4·3-s − 4·7-s + 6·9-s − 16-s − 4·19-s + 16·21-s + 4·27-s + 20·31-s − 4·37-s + 4·48-s + 8·49-s + 16·57-s + 32·61-s − 24·63-s + 20·67-s − 4·73-s − 40·79-s − 37·81-s − 80·93-s − 28·97-s − 4·109-s + 16·111-s + 4·112-s + 127-s + 131-s + 16·133-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.51·7-s + 2·9-s − 1/4·16-s − 0.917·19-s + 3.49·21-s + 0.769·27-s + 3.59·31-s − 0.657·37-s + 0.577·48-s + 8/7·49-s + 2.11·57-s + 4.09·61-s − 3.02·63-s + 2.44·67-s − 0.468·73-s − 4.50·79-s − 4.11·81-s − 8.29·93-s − 2.84·97-s − 0.383·109-s + 1.51·111-s + 0.377·112-s + 0.0887·127-s + 0.0873·131-s + 1.38·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3960065740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3960065740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2722 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 15518 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.83822947520880574577364654393, −7.59181066812645919498436541365, −7.14141003972742684763235584438, −6.75899751472245250071524594636, −6.75660616510535113707480455451, −6.74557801224362246443472253505, −6.53910000091238888954378893823, −6.16030917518269674007320590691, −5.85971717573553252982609541817, −5.69794730910731047127424763801, −5.53266345040913408084528041862, −5.25512262109101644874096447763, −4.94803908289710508373848657373, −4.60827072143380032191079711651, −4.42871004452897072754881790645, −4.09101466394133302183611782626, −3.86625601007857624888927256146, −3.49778355333802456078703988908, −2.86027029231990694911627953365, −2.73324232175376469880236674273, −2.66679232346809366379394973186, −1.99684177936964598146125445298, −1.28217490816909952836313307976, −0.825172695563514558197423061919, −0.35295455859912775921321378120,
0.35295455859912775921321378120, 0.825172695563514558197423061919, 1.28217490816909952836313307976, 1.99684177936964598146125445298, 2.66679232346809366379394973186, 2.73324232175376469880236674273, 2.86027029231990694911627953365, 3.49778355333802456078703988908, 3.86625601007857624888927256146, 4.09101466394133302183611782626, 4.42871004452897072754881790645, 4.60827072143380032191079711651, 4.94803908289710508373848657373, 5.25512262109101644874096447763, 5.53266345040913408084528041862, 5.69794730910731047127424763801, 5.85971717573553252982609541817, 6.16030917518269674007320590691, 6.53910000091238888954378893823, 6.74557801224362246443472253505, 6.75660616510535113707480455451, 6.75899751472245250071524594636, 7.14141003972742684763235584438, 7.59181066812645919498436541365, 7.83822947520880574577364654393
