| L(s) = 1 | − 2·4-s − 5·7-s − 3·13-s − 9·19-s − 10·25-s + 10·28-s − 15·31-s − 37-s + 5·43-s + 18·49-s + 6·52-s − 12·61-s + 8·64-s − 11·67-s − 27·73-s + 18·76-s + 13·79-s + 15·91-s + 24·97-s + 20·100-s + 17·109-s + 22·121-s + 30·124-s + 127-s + 131-s + 45·133-s + 137-s + ⋯ |
| L(s) = 1 | − 4-s − 1.88·7-s − 0.832·13-s − 2.06·19-s − 2·25-s + 1.88·28-s − 2.69·31-s − 0.164·37-s + 0.762·43-s + 18/7·49-s + 0.832·52-s − 1.53·61-s + 64-s − 1.34·67-s − 3.16·73-s + 2.06·76-s + 1.46·79-s + 1.57·91-s + 2.43·97-s + 2·100-s + 1.62·109-s + 2·121-s + 2.69·124-s + 0.0887·127-s + 0.0873·131-s + 3.90·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−10.24092751073910361096319906159, −10.20447428935442225403065759091, −9.413872586616404010807268202639, −9.384611413884364812123808257236, −8.752751614981544701371084311163, −8.723568286164489635619340751162, −7.60825397842531700891434162394, −7.54360581264375336559611290053, −6.93136601104677721359850135166, −6.34201324249135953880320648178, −5.78743003490096479476604164154, −5.75390207970056771012817440340, −4.63673663986755196207059217509, −4.43065648145326555356587999994, −3.67158954279706327328960078890, −3.46672728848904923553238531119, −2.46124965888072290614894590866, −1.93885469898278456073909423767, 0, 0,
1.93885469898278456073909423767, 2.46124965888072290614894590866, 3.46672728848904923553238531119, 3.67158954279706327328960078890, 4.43065648145326555356587999994, 4.63673663986755196207059217509, 5.75390207970056771012817440340, 5.78743003490096479476604164154, 6.34201324249135953880320648178, 6.93136601104677721359850135166, 7.54360581264375336559611290053, 7.60825397842531700891434162394, 8.723568286164489635619340751162, 8.752751614981544701371084311163, 9.384611413884364812123808257236, 9.413872586616404010807268202639, 10.20447428935442225403065759091, 10.24092751073910361096319906159
