L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s + 9-s − 2·10-s + 16-s − 10·17-s − 18-s + 2·20-s + 3·25-s − 2·29-s − 32-s + 10·34-s + 36-s − 12·37-s − 2·40-s + 12·41-s + 2·45-s + 2·49-s − 3·50-s + 4·53-s + 2·58-s + 64-s − 10·68-s − 72-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1/4·16-s − 2.42·17-s − 0.235·18-s + 0.447·20-s + 3/5·25-s − 0.371·29-s − 0.176·32-s + 1.71·34-s + 1/6·36-s − 1.97·37-s − 0.316·40-s + 1.87·41-s + 0.298·45-s + 2/7·49-s − 0.424·50-s + 0.549·53-s + 0.262·58-s + 1/8·64-s − 1.21·68-s − 0.117·72-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.72399680577781381068190941188, −7.29567887011859489397561277183, −7.00842308966653019977347141560, −6.55866549638180711725462165599, −6.06221011586077769044205992754, −5.84883861903278300730132626923, −5.03941182584389079435190834405, −4.77967253103443767951626263272, −4.11455840147798965310194787051, −3.64250590211989440321490015875, −2.82371961588642408668250267269, −2.23299012576811920414939385951, −1.97420693452990182879932696732, −1.12490882824850953950834997471, 0,
1.12490882824850953950834997471, 1.97420693452990182879932696732, 2.23299012576811920414939385951, 2.82371961588642408668250267269, 3.64250590211989440321490015875, 4.11455840147798965310194787051, 4.77967253103443767951626263272, 5.03941182584389079435190834405, 5.84883861903278300730132626923, 6.06221011586077769044205992754, 6.55866549638180711725462165599, 7.00842308966653019977347141560, 7.29567887011859489397561277183, 7.72399680577781381068190941188