L(s) = 1 | + 2·3-s + 6·7-s + 2·9-s + 12·21-s − 2·23-s + 6·27-s + 24·41-s − 18·43-s − 14·47-s + 18·49-s + 16·61-s + 12·63-s − 6·67-s − 4·69-s + 11·81-s + 22·83-s + 36·101-s + 18·103-s − 26·107-s + 22·121-s + 48·123-s + 127-s − 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 2.26·7-s + 2/3·9-s + 2.61·21-s − 0.417·23-s + 1.15·27-s + 3.74·41-s − 2.74·43-s − 2.04·47-s + 18/7·49-s + 2.04·61-s + 1.51·63-s − 0.733·67-s − 0.481·69-s + 11/9·81-s + 2.41·83-s + 3.58·101-s + 1.77·103-s − 2.51·107-s + 2·121-s + 4.32·123-s + 0.0887·127-s − 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.399593892\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.399593892\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−9.408763275244201971869349750839, −9.193695830727847441299908954251, −8.554274151530085979335989973428, −8.458307799155253732547424686946, −7.996998356327819580249694621112, −7.908630604460117896774675840808, −7.31014811781256623610243778469, −7.08379155667793190431818848662, −6.25004377847683493159872062077, −6.11636726984132640216053581543, −5.28459583820797802653985419818, −4.92200478730739720731155649436, −4.67419135441479636121059417838, −4.22093113621251028108449619760, −3.55042703931782608738695196740, −3.23156886543503515115445643165, −2.26194430081822144363307138131, −2.25784406623962292843365434031, −1.51821692617085132752827091846, −0.900959785717195410354393173937,
0.900959785717195410354393173937, 1.51821692617085132752827091846, 2.25784406623962292843365434031, 2.26194430081822144363307138131, 3.23156886543503515115445643165, 3.55042703931782608738695196740, 4.22093113621251028108449619760, 4.67419135441479636121059417838, 4.92200478730739720731155649436, 5.28459583820797802653985419818, 6.11636726984132640216053581543, 6.25004377847683493159872062077, 7.08379155667793190431818848662, 7.31014811781256623610243778469, 7.908630604460117896774675840808, 7.996998356327819580249694621112, 8.458307799155253732547424686946, 8.554274151530085979335989973428, 9.193695830727847441299908954251, 9.408763275244201971869349750839