| L(s) = 1 | + 2-s − 3-s − 6-s − 7-s − 8-s − 5·11-s − 14-s − 16-s − 4·17-s − 8·19-s + 21-s − 5·22-s − 4·23-s + 24-s + 27-s − 10·29-s − 3·31-s + 5·33-s − 4·34-s − 4·37-s − 8·38-s + 42-s − 4·43-s − 4·46-s − 6·47-s + 48-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 1.50·11-s − 0.267·14-s − 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.218·21-s − 1.06·22-s − 0.834·23-s + 0.204·24-s + 0.192·27-s − 1.85·29-s − 0.538·31-s + 0.870·33-s − 0.685·34-s − 0.657·37-s − 1.29·38-s + 0.154·42-s − 0.609·43-s − 0.589·46-s − 0.875·47-s + 0.144·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.678689101341295452822122358106, −9.434956655005391004339348037327, −8.810438674498531201628486466349, −8.493348126849116086106028590215, −7.86893710961468314708691687304, −7.82401896633040635985045540793, −6.97977074030033089473429676998, −6.51360396989113378177973354703, −6.41007476745326135587539873599, −5.76410473914077211476771835103, −5.21668572551171480318499492699, −5.13071125691891994709698534688, −4.50595392829906558172204785000, −3.78681309717082734740436612075, −3.73006254594142138554368954655, −2.78962558431298810407728049052, −2.25711565454085477630407995337, −1.78347108183832061810664122247, 0, 0,
1.78347108183832061810664122247, 2.25711565454085477630407995337, 2.78962558431298810407728049052, 3.73006254594142138554368954655, 3.78681309717082734740436612075, 4.50595392829906558172204785000, 5.13071125691891994709698534688, 5.21668572551171480318499492699, 5.76410473914077211476771835103, 6.41007476745326135587539873599, 6.51360396989113378177973354703, 6.97977074030033089473429676998, 7.82401896633040635985045540793, 7.86893710961468314708691687304, 8.493348126849116086106028590215, 8.810438674498531201628486466349, 9.434956655005391004339348037327, 9.678689101341295452822122358106
