L(s) = 1 | − 3-s + 5-s + 5·7-s + 11-s − 15-s + 8·17-s − 4·19-s − 5·21-s − 4·23-s + 5·25-s + 27-s + 10·29-s − 7·31-s − 33-s + 5·35-s + 8·37-s + 8·41-s − 20·43-s − 6·47-s + 18·49-s − 8·51-s − 53-s + 55-s + 4·57-s + 9·59-s − 2·61-s + 2·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.88·7-s + 0.301·11-s − 0.258·15-s + 1.94·17-s − 0.917·19-s − 1.09·21-s − 0.834·23-s + 25-s + 0.192·27-s + 1.85·29-s − 1.25·31-s − 0.174·33-s + 0.845·35-s + 1.31·37-s + 1.24·41-s − 3.04·43-s − 0.875·47-s + 18/7·49-s − 1.12·51-s − 0.137·53-s + 0.134·55-s + 0.529·57-s + 1.17·59-s − 0.256·61-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.998926650\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.998926650\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−9.771575565859570022491580385872, −9.659518403218010228369853799927, −8.742246299390820731005239613668, −8.648337938869299425675133949218, −8.064522442701276376092597899293, −8.022795475321647779525766668366, −7.38342539216342759655630631574, −7.00829318203567106109216455654, −6.36985627647395589409330629028, −6.06543630531402533596344589126, −5.68887996800110533036229011674, −5.05290141194100084625959097186, −4.77944480205365351660651605344, −4.63989452383242343099588936017, −3.64120534326883049856286020436, −3.46389469796025996116386638006, −2.45067608633037747285680715120, −2.05058978641858857343839272505, −1.32647461576426298406969447795, −0.846239260396293073982028686291,
0.846239260396293073982028686291, 1.32647461576426298406969447795, 2.05058978641858857343839272505, 2.45067608633037747285680715120, 3.46389469796025996116386638006, 3.64120534326883049856286020436, 4.63989452383242343099588936017, 4.77944480205365351660651605344, 5.05290141194100084625959097186, 5.68887996800110533036229011674, 6.06543630531402533596344589126, 6.36985627647395589409330629028, 7.00829318203567106109216455654, 7.38342539216342759655630631574, 8.022795475321647779525766668366, 8.064522442701276376092597899293, 8.648337938869299425675133949218, 8.742246299390820731005239613668, 9.659518403218010228369853799927, 9.771575565859570022491580385872