| L(s) = 1 | − 2-s + 2·3-s − 4-s − 2·6-s + 3·8-s + 3·9-s + 8·11-s − 2·12-s − 16-s + 4·17-s − 3·18-s − 8·19-s − 8·22-s + 6·24-s + 25-s + 4·27-s − 5·32-s + 16·33-s − 4·34-s − 3·36-s + 8·38-s − 12·41-s − 8·43-s − 8·44-s − 2·48-s − 14·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.816·6-s + 1.06·8-s + 9-s + 2.41·11-s − 0.577·12-s − 1/4·16-s + 0.970·17-s − 0.707·18-s − 1.83·19-s − 1.70·22-s + 1.22·24-s + 1/5·25-s + 0.769·27-s − 0.883·32-s + 2.78·33-s − 0.685·34-s − 1/2·36-s + 1.29·38-s − 1.87·41-s − 1.21·43-s − 1.20·44-s − 0.288·48-s − 2·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−7.66093767427808108694408525270, −7.26587220658258987619659124673, −6.48654966834348628333399350819, −6.38045586811398028783284364586, −6.12557729231289131795228792800, −4.88964690901686673600763522174, −4.88823595164803979266926983864, −4.35289952326547862812898780702, −3.68764508509381599036667118399, −3.56799192480000330057521215275, −3.04823786935056050267442941329, −2.05348697617991381387369600911, −1.53827946418072491307494025034, −1.32104408130134713650580275933, 0,
1.32104408130134713650580275933, 1.53827946418072491307494025034, 2.05348697617991381387369600911, 3.04823786935056050267442941329, 3.56799192480000330057521215275, 3.68764508509381599036667118399, 4.35289952326547862812898780702, 4.88823595164803979266926983864, 4.88964690901686673600763522174, 6.12557729231289131795228792800, 6.38045586811398028783284364586, 6.48654966834348628333399350819, 7.26587220658258987619659124673, 7.66093767427808108694408525270
