| L(s) = 1 | + 2·2-s + 2·4-s + 4·7-s − 9-s + 8·14-s − 4·16-s + 4·17-s − 2·18-s − 8·23-s + 8·28-s + 4·31-s − 8·32-s + 8·34-s − 2·36-s + 4·41-s − 16·46-s + 24·47-s − 2·49-s + 8·62-s − 4·63-s − 8·64-s + 8·68-s + 24·71-s + 12·73-s + 20·79-s + 81-s + 8·82-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.51·7-s − 1/3·9-s + 2.13·14-s − 16-s + 0.970·17-s − 0.471·18-s − 1.66·23-s + 1.51·28-s + 0.718·31-s − 1.41·32-s + 1.37·34-s − 1/3·36-s + 0.624·41-s − 2.35·46-s + 3.50·47-s − 2/7·49-s + 1.01·62-s − 0.503·63-s − 64-s + 0.970·68-s + 2.84·71-s + 1.40·73-s + 2.25·79-s + 1/9·81-s + 0.883·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.479250433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.479250433\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23500814641734376313391646622, −10.65287304210731381463935127440, −10.06659894742228226185856138012, −9.679767950589030398625343386252, −9.101765155209523662115343299040, −8.563930866421553393474533315776, −8.186683173977034577071938201466, −7.66074180264381018851565734782, −7.48275774063026117609266718638, −6.56558913896775867155940768459, −6.27234888394639902856459079945, −5.53490213627277680282315924765, −5.48927151613527267240312624642, −4.78878848102329281462321447025, −4.41710692676585715941599052066, −3.79255416557036208511651032831, −3.44211313188152667594043225159, −2.34544413746819339102586438578, −2.22461477781248693175524155381, −1.00492182132539981994304661685,
1.00492182132539981994304661685, 2.22461477781248693175524155381, 2.34544413746819339102586438578, 3.44211313188152667594043225159, 3.79255416557036208511651032831, 4.41710692676585715941599052066, 4.78878848102329281462321447025, 5.48927151613527267240312624642, 5.53490213627277680282315924765, 6.27234888394639902856459079945, 6.56558913896775867155940768459, 7.48275774063026117609266718638, 7.66074180264381018851565734782, 8.186683173977034577071938201466, 8.563930866421553393474533315776, 9.101765155209523662115343299040, 9.679767950589030398625343386252, 10.06659894742228226185856138012, 10.65287304210731381463935127440, 11.23500814641734376313391646622
