| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 8-s + 3·11-s + 5·13-s − 14-s − 16-s + 3·17-s − 5·19-s + 21-s + 3·22-s + 6·23-s + 24-s − 10·25-s + 5·26-s + 27-s + 9·29-s + 16·31-s − 3·33-s + 3·34-s − 8·37-s − 5·38-s − 5·39-s − 3·41-s + 42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 1.14·19-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 2·25-s + 0.980·26-s + 0.192·27-s + 1.67·29-s + 2.87·31-s − 0.522·33-s + 0.514·34-s − 1.31·37-s − 0.811·38-s − 0.800·39-s − 0.468·41-s + 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.048584117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.048584117\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−11.27467675444554506627348569976, −10.43665446237303050423048562125, −10.17921788330878460074369981724, −10.02605263889822091944870710653, −9.122340129898367408359917653256, −8.862927880834299839115338554495, −8.218760440740126732519333153441, −8.206461721416714192763360892477, −7.20688672234468133435500442623, −6.57202235152663544342250571000, −6.50643086392239607624689136621, −5.93282311916951321499209389756, −5.62708270514104307497978292971, −4.70971873196188448076294038605, −4.56427042930626092281714506410, −3.82168485186125513243235329820, −3.36605741552866436779474637925, −2.78155526401293814983481276927, −1.69814671503214991208220847581, −0.826149236098139701427696371742,
0.826149236098139701427696371742, 1.69814671503214991208220847581, 2.78155526401293814983481276927, 3.36605741552866436779474637925, 3.82168485186125513243235329820, 4.56427042930626092281714506410, 4.70971873196188448076294038605, 5.62708270514104307497978292971, 5.93282311916951321499209389756, 6.50643086392239607624689136621, 6.57202235152663544342250571000, 7.20688672234468133435500442623, 8.206461721416714192763360892477, 8.218760440740126732519333153441, 8.862927880834299839115338554495, 9.122340129898367408359917653256, 10.02605263889822091944870710653, 10.17921788330878460074369981724, 10.43665446237303050423048562125, 11.27467675444554506627348569976
