| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·12-s + 16-s − 8·19-s + 25-s − 4·27-s − 2·31-s + 36-s − 16·37-s + 4·43-s + 2·48-s − 14·49-s − 16·57-s + 64-s + 8·67-s + 12·73-s + 2·75-s − 8·76-s − 8·79-s − 11·81-s − 4·93-s − 4·97-s + 100-s + 16·103-s − 4·108-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 1/4·16-s − 1.83·19-s + 1/5·25-s − 0.769·27-s − 0.359·31-s + 1/6·36-s − 2.63·37-s + 0.609·43-s + 0.288·48-s − 2·49-s − 2.11·57-s + 1/8·64-s + 0.977·67-s + 1.40·73-s + 0.230·75-s − 0.917·76-s − 0.900·79-s − 1.22·81-s − 0.414·93-s − 0.406·97-s + 1/10·100-s + 1.57·103-s − 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.915375431544173023936891872650, −7.82324426194698016701524671113, −6.90360610726231452864919251865, −6.85566084303057349352756596641, −6.38928498895932353355711112534, −5.66689224147231460184537903057, −5.36013908964288749051473827734, −4.62364450878552237809302992665, −4.18447072325511001626508694490, −3.41908351741315490904238557756, −3.35246501897724533562865353410, −2.44569227539742541309238526143, −2.09679879971450421520638196039, −1.50002343442867035876074318730, 0,
1.50002343442867035876074318730, 2.09679879971450421520638196039, 2.44569227539742541309238526143, 3.35246501897724533562865353410, 3.41908351741315490904238557756, 4.18447072325511001626508694490, 4.62364450878552237809302992665, 5.36013908964288749051473827734, 5.66689224147231460184537903057, 6.38928498895932353355711112534, 6.85566084303057349352756596641, 6.90360610726231452864919251865, 7.82324426194698016701524671113, 7.915375431544173023936891872650
