| L(s) = 1 | + 3·2-s + 3·4-s − 3·8-s + 2·9-s + 3·11-s − 13·16-s + 6·18-s + 9·22-s + 5·23-s + 2·25-s + 29-s − 15·32-s + 6·36-s + 5·37-s − 4·43-s + 9·44-s + 15·46-s + 6·50-s + 7·53-s + 3·58-s + 3·64-s + 10·67-s + 3·71-s − 6·72-s + 15·74-s − 19·79-s − 5·81-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.06·8-s + 2/3·9-s + 0.904·11-s − 3.25·16-s + 1.41·18-s + 1.91·22-s + 1.04·23-s + 2/5·25-s + 0.185·29-s − 2.65·32-s + 36-s + 0.821·37-s − 0.609·43-s + 1.35·44-s + 2.21·46-s + 0.848·50-s + 0.961·53-s + 0.393·58-s + 3/8·64-s + 1.22·67-s + 0.356·71-s − 0.707·72-s + 1.74·74-s − 2.13·79-s − 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 463393 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 463393 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.023028742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.023028742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 193 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 15 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
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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
−8.555514603686128420147215346613, −8.192533575352427375829956920884, −7.31286984088128030033966998939, −6.90598756340387469643324019411, −6.65434687909378561617183147912, −5.94189203902399198886555953773, −5.70205846503602714193238169574, −5.12218155653346935244606746242, −4.55049610840503439423130950155, −4.40241956155225358676922519100, −3.82515546834938134075524046721, −3.28779914663384711400305876990, −2.87384812911568221380280810280, −2.00420403481848234929348945730, −0.902535160792512842820849355157,
0.902535160792512842820849355157, 2.00420403481848234929348945730, 2.87384812911568221380280810280, 3.28779914663384711400305876990, 3.82515546834938134075524046721, 4.40241956155225358676922519100, 4.55049610840503439423130950155, 5.12218155653346935244606746242, 5.70205846503602714193238169574, 5.94189203902399198886555953773, 6.65434687909378561617183147912, 6.90598756340387469643324019411, 7.31286984088128030033966998939, 8.192533575352427375829956920884, 8.555514603686128420147215346613
