| L(s) = 1 | − 3-s + 4-s − 3·7-s − 2·9-s − 12-s − 3·13-s + 16-s − 10·19-s + 3·21-s + 7·25-s + 5·27-s − 3·28-s + 31-s − 2·36-s − 20·37-s + 3·39-s + 3·43-s − 48-s − 7·49-s − 3·52-s + 10·57-s − 5·61-s + 6·63-s + 64-s + 7·73-s − 7·75-s − 10·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.13·7-s − 2/3·9-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 2.29·19-s + 0.654·21-s + 7/5·25-s + 0.962·27-s − 0.566·28-s + 0.179·31-s − 1/3·36-s − 3.28·37-s + 0.480·39-s + 0.457·43-s − 0.144·48-s − 49-s − 0.416·52-s + 1.32·57-s − 0.640·61-s + 0.755·63-s + 1/8·64-s + 0.819·73-s − 0.808·75-s − 1.14·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−10.31921078068295268954735015653, −9.868558809720640429553609509893, −9.126146945252641906700520165591, −8.594669133476250084003395910980, −8.293018407324021441761316366350, −7.23652540018263229315188777051, −6.83348980079870200421862284064, −6.42040229058336829772160771828, −5.92508305808422035516780366968, −5.12440353247235808100972100234, −4.60958026920970568236320356151, −3.54362019383048875451988339218, −2.92161708698637847242961500221, −2.03868503630672899570387548728, 0,
2.03868503630672899570387548728, 2.92161708698637847242961500221, 3.54362019383048875451988339218, 4.60958026920970568236320356151, 5.12440353247235808100972100234, 5.92508305808422035516780366968, 6.42040229058336829772160771828, 6.83348980079870200421862284064, 7.23652540018263229315188777051, 8.293018407324021441761316366350, 8.594669133476250084003395910980, 9.126146945252641906700520165591, 9.868558809720640429553609509893, 10.31921078068295268954735015653
