| L(s) = 1 | − 2·3-s − 7-s + 9-s + 2·13-s + 8·19-s + 2·21-s + 6·25-s + 4·27-s − 4·39-s − 6·43-s − 6·49-s − 16·57-s − 63-s − 24·67-s + 3·73-s − 12·75-s + 4·79-s − 11·81-s − 2·91-s + 16·97-s − 20·103-s − 20·109-s + 2·117-s − 14·121-s + 127-s + 12·129-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.83·19-s + 0.436·21-s + 6/5·25-s + 0.769·27-s − 0.640·39-s − 0.914·43-s − 6/7·49-s − 2.11·57-s − 0.125·63-s − 2.93·67-s + 0.351·73-s − 1.38·75-s + 0.450·79-s − 1.22·81-s − 0.209·91-s + 1.62·97-s − 1.97·103-s − 1.91·109-s + 0.184·117-s − 1.27·121-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1177344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.71968581557394793218861826133, −7.37922588907965065968457258226, −6.68490883990873713700390568289, −6.55759546834720038470940418823, −6.10494117091431111238007847241, −5.52365210534089094926857857790, −5.18697241857922415537295873734, −4.89867094903025985609584882410, −4.22812211466913323241727072154, −3.62586563081787721634660112685, −3.03000411373336209869450724782, −2.73300534630563289343863580354, −1.51852450521011498566722278099, −1.05994469783971410761088492203, 0,
1.05994469783971410761088492203, 1.51852450521011498566722278099, 2.73300534630563289343863580354, 3.03000411373336209869450724782, 3.62586563081787721634660112685, 4.22812211466913323241727072154, 4.89867094903025985609584882410, 5.18697241857922415537295873734, 5.52365210534089094926857857790, 6.10494117091431111238007847241, 6.55759546834720038470940418823, 6.68490883990873713700390568289, 7.37922588907965065968457258226, 7.71968581557394793218861826133
