| L(s) = 1 | − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s + 2·13-s + 16-s − 4·21-s − 25-s + 4·27-s + 2·28-s − 8·31-s + 36-s + 2·37-s − 4·39-s − 14·43-s − 2·48-s − 10·49-s + 2·52-s − 12·61-s + 2·63-s + 64-s − 12·67-s − 32·73-s + 2·75-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.872·21-s − 1/5·25-s + 0.769·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s + 0.328·37-s − 0.640·39-s − 2.13·43-s − 0.288·48-s − 1.42·49-s + 0.277·52-s − 1.53·61-s + 0.251·63-s + 1/8·64-s − 1.46·67-s − 3.74·73-s + 0.230·75-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.634946951138828134245935269889, −8.034724118742405492983809868289, −7.44127521940938968208670733555, −7.24093067434887596393144130153, −6.47089636707471054088078851688, −6.16939419963925115717727014562, −5.76247129124407646948114093200, −5.20473693200457193183888158657, −4.74697441275744468656593386099, −4.28339974110992017605243136171, −3.38939971276358052816888131275, −2.96367359035443357572862250415, −1.82358609795462761142951667346, −1.41901056582032844465265399635, 0,
1.41901056582032844465265399635, 1.82358609795462761142951667346, 2.96367359035443357572862250415, 3.38939971276358052816888131275, 4.28339974110992017605243136171, 4.74697441275744468656593386099, 5.20473693200457193183888158657, 5.76247129124407646948114093200, 6.16939419963925115717727014562, 6.47089636707471054088078851688, 7.24093067434887596393144130153, 7.44127521940938968208670733555, 8.034724118742405492983809868289, 8.634946951138828134245935269889
