| L(s) = 1 | − 3-s + 9-s + 11-s + 9·17-s + 11·19-s − 6·25-s − 27-s − 33-s + 10·41-s + 7·43-s − 49-s − 9·51-s − 11·57-s − 13·59-s + 19·67-s − 3·73-s + 6·75-s + 81-s + 9·83-s + 8·89-s − 4·97-s + 99-s + 23·107-s + 22·113-s − 19·121-s − 10·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s + 2.18·17-s + 2.52·19-s − 6/5·25-s − 0.192·27-s − 0.174·33-s + 1.56·41-s + 1.06·43-s − 1/7·49-s − 1.26·51-s − 1.45·57-s − 1.69·59-s + 2.32·67-s − 0.351·73-s + 0.692·75-s + 1/9·81-s + 0.987·83-s + 0.847·89-s − 0.406·97-s + 0.100·99-s + 2.22·107-s + 2.06·113-s − 1.72·121-s − 0.901·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1492992 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.318505206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.318505206\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.82598985117731028002438569695, −7.45467202228909310283112342895, −7.28973930528926467090609933874, −6.53358840827559420948845495911, −6.00559210319755557479630670628, −5.74212141194382102798792858264, −5.38340550640765754654281610127, −4.93624589487615704829879746125, −4.38297900779946630953865967196, −3.62098592791310876617873286448, −3.47664260544686058202918988295, −2.85354914494671655649933168606, −2.05472103840942285504187062292, −1.18124747024371101469915651399, −0.826887052571451758646765985482,
0.826887052571451758646765985482, 1.18124747024371101469915651399, 2.05472103840942285504187062292, 2.85354914494671655649933168606, 3.47664260544686058202918988295, 3.62098592791310876617873286448, 4.38297900779946630953865967196, 4.93624589487615704829879746125, 5.38340550640765754654281610127, 5.74212141194382102798792858264, 6.00559210319755557479630670628, 6.53358840827559420948845495911, 7.28973930528926467090609933874, 7.45467202228909310283112342895, 7.82598985117731028002438569695
