| L(s) = 1 | − 2·3-s + 4·5-s + 6·7-s + 2·9-s − 4·11-s + 6·13-s − 8·15-s + 2·17-s − 12·21-s + 2·23-s + 11·25-s − 6·27-s + 8·33-s + 24·35-s − 2·37-s − 12·39-s + 20·41-s + 10·43-s + 8·45-s − 6·47-s + 18·49-s − 4·51-s + 10·53-s − 16·55-s + 12·63-s + 24·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 2.26·7-s + 2/3·9-s − 1.20·11-s + 1.66·13-s − 2.06·15-s + 0.485·17-s − 2.61·21-s + 0.417·23-s + 11/5·25-s − 1.15·27-s + 1.39·33-s + 4.05·35-s − 0.328·37-s − 1.92·39-s + 3.12·41-s + 1.52·43-s + 1.19·45-s − 0.875·47-s + 18/7·49-s − 0.560·51-s + 1.37·53-s − 2.15·55-s + 1.51·63-s + 2.97·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.469760858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.469760858\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.922688579981479137852418893454, −9.549856367742655625880538065654, −8.970314845408176973456245845473, −8.820400159592285487358857898541, −8.038188761670882101439840911437, −8.017271231326481780309698016654, −7.40362973010315184315206886095, −7.03880615033601867614165264792, −6.21056273596991008821006127045, −6.03378269273264795310538736371, −5.53628494050908219257323427391, −5.49717969951708368035730331488, −4.97867935868452473061693244619, −4.49410541001700249412197387359, −4.06396034048631165250022528978, −3.20007882476140098809173043417, −2.25433379753719316251552634035, −2.19173059087105296127762974667, −1.14858858475842360909772750475, −1.10858717082779186771415028761,
1.10858717082779186771415028761, 1.14858858475842360909772750475, 2.19173059087105296127762974667, 2.25433379753719316251552634035, 3.20007882476140098809173043417, 4.06396034048631165250022528978, 4.49410541001700249412197387359, 4.97867935868452473061693244619, 5.49717969951708368035730331488, 5.53628494050908219257323427391, 6.03378269273264795310538736371, 6.21056273596991008821006127045, 7.03880615033601867614165264792, 7.40362973010315184315206886095, 8.017271231326481780309698016654, 8.038188761670882101439840911437, 8.820400159592285487358857898541, 8.970314845408176973456245845473, 9.549856367742655625880538065654, 9.922688579981479137852418893454
