| L(s) = 1 | − 2·3-s − 2·5-s − 4·7-s + 6·11-s + 2·13-s + 4·15-s + 2·19-s + 8·21-s − 6·23-s + 3·25-s + 2·27-s − 12·29-s − 10·31-s − 12·33-s + 8·35-s − 8·37-s − 4·39-s − 10·43-s − 12·47-s − 2·49-s − 12·55-s − 4·57-s + 6·59-s + 4·61-s − 4·65-s + 8·67-s + 12·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1.51·7-s + 1.80·11-s + 0.554·13-s + 1.03·15-s + 0.458·19-s + 1.74·21-s − 1.25·23-s + 3/5·25-s + 0.384·27-s − 2.22·29-s − 1.79·31-s − 2.08·33-s + 1.35·35-s − 1.31·37-s − 0.640·39-s − 1.52·43-s − 1.75·47-s − 2/7·49-s − 1.61·55-s − 0.529·57-s + 0.781·59-s + 0.512·61-s − 0.496·65-s + 0.977·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.532663191880723444619536247441, −9.520396530595932899143950961913, −8.967206648728870014215894131024, −8.540717878316464630724059987299, −8.085339355611054246272212776661, −7.49276571100856719837184004871, −6.96712216720779269762232073081, −6.79838215939637904678030754983, −6.15940545510306890101837070572, −6.12996587625445438988118560282, −5.32936653089787902577483318484, −5.24556846747375416793353123411, −4.25980771066364625189641702523, −3.81607625168915196341785738898, −3.41420358823914627451734117820, −3.33321581223917908753820993425, −1.98037084213370900213557926370, −1.41141287518027260408649480549, 0, 0,
1.41141287518027260408649480549, 1.98037084213370900213557926370, 3.33321581223917908753820993425, 3.41420358823914627451734117820, 3.81607625168915196341785738898, 4.25980771066364625189641702523, 5.24556846747375416793353123411, 5.32936653089787902577483318484, 6.12996587625445438988118560282, 6.15940545510306890101837070572, 6.79838215939637904678030754983, 6.96712216720779269762232073081, 7.49276571100856719837184004871, 8.085339355611054246272212776661, 8.540717878316464630724059987299, 8.967206648728870014215894131024, 9.520396530595932899143950961913, 9.532663191880723444619536247441
