| L(s) = 1 | + 2·3-s + 4·7-s + 9-s − 4·13-s + 8·19-s + 8·21-s + 25-s − 4·27-s − 8·31-s − 4·37-s − 8·39-s + 20·43-s − 2·49-s + 16·57-s − 4·61-s + 4·63-s − 4·67-s + 4·73-s + 2·75-s + 16·79-s − 11·81-s − 16·91-s − 16·93-s + 4·97-s + 28·103-s − 4·109-s − 8·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.10·13-s + 1.83·19-s + 1.74·21-s + 1/5·25-s − 0.769·27-s − 1.43·31-s − 0.657·37-s − 1.28·39-s + 3.04·43-s − 2/7·49-s + 2.11·57-s − 0.512·61-s + 0.503·63-s − 0.488·67-s + 0.468·73-s + 0.230·75-s + 1.80·79-s − 1.22·81-s − 1.67·91-s − 1.65·93-s + 0.406·97-s + 2.75·103-s − 0.383·109-s − 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.708376006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.708376006\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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
−7.87489459574230399033328236522, −7.79407539110457648656273831410, −7.60753011721166622675464955174, −7.09464940360169401329405651291, −6.54046274838058996502273479449, −5.59481538925271546647259599831, −5.52947923104725115211828893156, −4.97997261589161680655107982307, −4.45194901772205251160909130654, −4.00111153404090288423211291284, −3.18925682979485723001966009201, −3.01436611750632173441455711709, −2.04591517438255900406864260966, −1.91202694614300604029027352933, −0.862970868573876613686429334418,
0.862970868573876613686429334418, 1.91202694614300604029027352933, 2.04591517438255900406864260966, 3.01436611750632173441455711709, 3.18925682979485723001966009201, 4.00111153404090288423211291284, 4.45194901772205251160909130654, 4.97997261589161680655107982307, 5.52947923104725115211828893156, 5.59481538925271546647259599831, 6.54046274838058996502273479449, 7.09464940360169401329405651291, 7.60753011721166622675464955174, 7.79407539110457648656273831410, 7.87489459574230399033328236522
