| L(s) = 1 | − 2-s − 2·5-s + 7-s + 8-s + 2·10-s − 4·11-s − 7·13-s − 14-s − 16-s + 4·17-s + 10·19-s + 4·22-s + 2·23-s + 5·25-s + 7·26-s − 8·29-s − 4·34-s − 2·35-s − 7·37-s − 10·38-s − 2·40-s − 10·41-s − 7·43-s − 2·46-s − 12·47-s − 6·49-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.894·5-s + 0.377·7-s + 0.353·8-s + 0.632·10-s − 1.20·11-s − 1.94·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s + 2.29·19-s + 0.852·22-s + 0.417·23-s + 25-s + 1.37·26-s − 1.48·29-s − 0.685·34-s − 0.338·35-s − 1.15·37-s − 1.62·38-s − 0.316·40-s − 1.56·41-s − 1.06·43-s − 0.294·46-s − 1.75·47-s − 6/7·49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.104742369509363342611921788856, −8.864313035883125917548251353899, −8.257326285740225326049895386725, −7.87585738921994003510135133068, −7.68999835169960255030943630466, −7.41239941253462153077272229606, −6.95264049494777236795641668260, −6.74366515331278441124043866494, −5.62654649462887656807928878323, −5.37927584027098738549325924526, −5.16999632046475785703671008390, −4.68278942564400191380761472211, −4.28104635111426444135302824762, −3.35807897205814353348246527763, −3.02046316028454503640250051689, −2.90337906341847165197351230257, −1.63218485623769897330193163306, −1.48536303422127790427842873654, 0, 0,
1.48536303422127790427842873654, 1.63218485623769897330193163306, 2.90337906341847165197351230257, 3.02046316028454503640250051689, 3.35807897205814353348246527763, 4.28104635111426444135302824762, 4.68278942564400191380761472211, 5.16999632046475785703671008390, 5.37927584027098738549325924526, 5.62654649462887656807928878323, 6.74366515331278441124043866494, 6.95264049494777236795641668260, 7.41239941253462153077272229606, 7.68999835169960255030943630466, 7.87585738921994003510135133068, 8.257326285740225326049895386725, 8.864313035883125917548251353899, 9.104742369509363342611921788856
