| L(s) = 1 | − 2·5-s + 4·11-s − 4·13-s + 2·17-s + 4·19-s − 8·23-s + 5·25-s − 12·29-s − 8·31-s − 6·37-s + 12·41-s + 8·43-s − 2·53-s − 8·55-s + 4·59-s + 2·61-s + 8·65-s + 4·67-s − 16·71-s − 10·73-s + 8·79-s + 8·83-s − 4·85-s − 6·89-s − 8·95-s + 4·97-s − 18·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 25-s − 2.22·29-s − 1.43·31-s − 0.986·37-s + 1.87·41-s + 1.21·43-s − 0.274·53-s − 1.07·55-s + 0.520·59-s + 0.256·61-s + 0.992·65-s + 0.488·67-s − 1.89·71-s − 1.17·73-s + 0.900·79-s + 0.878·83-s − 0.433·85-s − 0.635·89-s − 0.820·95-s + 0.406·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5412865254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5412865254\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.204046596487589180186904514203, −8.272553344426047313612471424621, −7.85512773338797021093032314339, −7.59683149763123796779310282940, −7.45974219323980043518733786024, −6.90630717232796421988368871476, −6.76247297475051832343364194177, −5.96531850397358392662028126581, −5.84162205342835299652013332632, −5.24370443692848804349558117651, −5.16727025592672306048199659099, −4.25537204051607515531005631063, −4.14836954244510441850702376959, −3.66051715115320171014303449147, −3.56105981485770482564089602739, −2.57539903730649162076855502659, −2.51177880333869842261581254860, −1.54596178224669240159256437825, −1.31174783387325350265933982568, −0.22571247862950536842039322377,
0.22571247862950536842039322377, 1.31174783387325350265933982568, 1.54596178224669240159256437825, 2.51177880333869842261581254860, 2.57539903730649162076855502659, 3.56105981485770482564089602739, 3.66051715115320171014303449147, 4.14836954244510441850702376959, 4.25537204051607515531005631063, 5.16727025592672306048199659099, 5.24370443692848804349558117651, 5.84162205342835299652013332632, 5.96531850397358392662028126581, 6.76247297475051832343364194177, 6.90630717232796421988368871476, 7.45974219323980043518733786024, 7.59683149763123796779310282940, 7.85512773338797021093032314339, 8.272553344426047313612471424621, 9.204046596487589180186904514203
