| L(s) = 1 | + 3-s − 4-s − 2·7-s + 9-s − 11-s − 12-s + 13-s − 3·16-s − 19-s − 2·21-s + 3·23-s − 5·25-s + 4·27-s + 2·28-s − 12·29-s − 2·31-s − 33-s − 36-s + 9·37-s + 39-s + 9·41-s + 8·43-s + 44-s + 6·47-s − 3·48-s − 2·49-s − 52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 3/4·16-s − 0.229·19-s − 0.436·21-s + 0.625·23-s − 25-s + 0.769·27-s + 0.377·28-s − 2.22·29-s − 0.359·31-s − 0.174·33-s − 1/6·36-s + 1.47·37-s + 0.160·39-s + 1.40·41-s + 1.21·43-s + 0.150·44-s + 0.875·47-s − 0.433·48-s − 2/7·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7085326897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7085326897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 76 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−18.5163110661, −17.9204703320, −17.5047916968, −16.7590473906, −16.1361499459, −15.8963922898, −15.1388896455, −14.6305506954, −14.1214851939, −13.3792820041, −12.9829072431, −12.7807060278, −11.7245782299, −11.1120503181, −10.5189949265, −9.59239201214, −9.28594284675, −8.78537555605, −7.76378213597, −7.35301661543, −6.32008887030, −5.61447907119, −4.44780824229, −3.70640401324, −2.48162258572,
2.48162258572, 3.70640401324, 4.44780824229, 5.61447907119, 6.32008887030, 7.35301661543, 7.76378213597, 8.78537555605, 9.28594284675, 9.59239201214, 10.5189949265, 11.1120503181, 11.7245782299, 12.7807060278, 12.9829072431, 13.3792820041, 14.1214851939, 14.6305506954, 15.1388896455, 15.8963922898, 16.1361499459, 16.7590473906, 17.5047916968, 17.9204703320, 18.5163110661
