L(s) = 1 | + 3·3-s + 4-s + 6·9-s + 4·11-s + 3·12-s + 16-s + 8·23-s + 9·27-s − 11·31-s + 12·33-s + 6·36-s + 10·37-s + 4·44-s + 3·48-s − 9·53-s + 4·59-s + 64-s − 67-s + 24·69-s − 7·71-s + 9·81-s + 5·89-s + 8·92-s − 33·93-s + 10·97-s + 24·99-s + 7·103-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1/2·4-s + 2·9-s + 1.20·11-s + 0.866·12-s + 1/4·16-s + 1.66·23-s + 1.73·27-s − 1.97·31-s + 2.08·33-s + 36-s + 1.64·37-s + 0.603·44-s + 0.433·48-s − 1.23·53-s + 0.520·59-s + 1/8·64-s − 0.122·67-s + 2.88·69-s − 0.830·71-s + 81-s + 0.529·89-s + 0.834·92-s − 3.42·93-s + 1.01·97-s + 2.41·99-s + 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.586995439\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.586995439\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.63999577629526656165496225751, −7.28737150535736495265852388819, −6.87053275673706509122526417135, −6.57023735729267708708480220041, −6.01940271920955655332286306650, −5.53188103062345586168768670390, −4.91293730733284943726912781833, −4.40823162701739063945950734759, −4.00660372708792656301351299546, −3.46494208996146654158605689096, −3.15851507863791526147362120409, −2.67734437969210087104606561283, −2.02854344382326446038850439984, −1.59751872240430127999078632900, −0.936202734408333832990538750936,
0.936202734408333832990538750936, 1.59751872240430127999078632900, 2.02854344382326446038850439984, 2.67734437969210087104606561283, 3.15851507863791526147362120409, 3.46494208996146654158605689096, 4.00660372708792656301351299546, 4.40823162701739063945950734759, 4.91293730733284943726912781833, 5.53188103062345586168768670390, 6.01940271920955655332286306650, 6.57023735729267708708480220041, 6.87053275673706509122526417135, 7.28737150535736495265852388819, 7.63999577629526656165496225751