L(s) = 1 | − 3·3-s − 2·5-s − 5·7-s + 6·9-s − 4·11-s − 3·13-s + 6·15-s − 7·17-s − 5·19-s + 15·21-s − 4·23-s + 5·25-s − 9·27-s + 29-s − 6·31-s + 12·33-s + 10·35-s − 11·37-s + 9·39-s + 9·41-s − 5·43-s − 12·45-s + 6·47-s + 18·49-s + 21·51-s − 3·53-s + 8·55-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 0.894·5-s − 1.88·7-s + 2·9-s − 1.20·11-s − 0.832·13-s + 1.54·15-s − 1.69·17-s − 1.14·19-s + 3.27·21-s − 0.834·23-s + 25-s − 1.73·27-s + 0.185·29-s − 1.07·31-s + 2.08·33-s + 1.69·35-s − 1.80·37-s + 1.44·39-s + 1.40·41-s − 0.762·43-s − 1.78·45-s + 0.875·47-s + 18/7·49-s + 2.94·51-s − 0.412·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3 T - 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−11.91531417605403419779691848026, −11.23687913171523506384582319772, −10.83591144998887698239228116061, −10.55689329101202657812793391340, −10.05121210160052439283654383345, −9.662878219913839207319557658412, −8.850537160039160845037015473376, −8.552443764793535707587050143900, −7.47382598407465557345766399459, −7.29112617787487007250299519865, −6.55768890581043077752145308703, −6.46454446132880625459248694753, −5.64055223211726024280230055273, −5.21690857840584288507882888627, −4.35617907475976882257230554471, −4.10171835893774217139521115615, −3.08224443917003121759885613474, −2.23495678738140070114362636140, 0, 0,
2.23495678738140070114362636140, 3.08224443917003121759885613474, 4.10171835893774217139521115615, 4.35617907475976882257230554471, 5.21690857840584288507882888627, 5.64055223211726024280230055273, 6.46454446132880625459248694753, 6.55768890581043077752145308703, 7.29112617787487007250299519865, 7.47382598407465557345766399459, 8.552443764793535707587050143900, 8.850537160039160845037015473376, 9.662878219913839207319557658412, 10.05121210160052439283654383345, 10.55689329101202657812793391340, 10.83591144998887698239228116061, 11.23687913171523506384582319772, 11.91531417605403419779691848026