| L(s) = 1 | − 4-s − 4·7-s + 16-s − 25-s + 4·28-s + 4·31-s + 10·49-s − 64-s + 4·73-s + 100-s − 4·112-s − 121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s − 4·7-s + 16-s − 25-s + 4·28-s + 4·31-s + 10·49-s − 64-s + 4·73-s + 100-s − 4·112-s − 121-s − 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 4·175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5428506120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5428506120\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.812982482223015601622394317940, −8.635439420297348243659689296025, −8.069197698724880143985706539150, −7.86335525327547243594087859369, −7.23033937902870630758178903027, −6.76507104220192588980680516194, −6.58207218974627329522711917673, −6.28366043460594445186935420235, −6.02359901027962498270680217781, −5.58971411688820460309784520000, −5.18832742479788427609666569673, −4.37328754425237463248456331961, −4.32884756750620426229543927381, −3.71259277831989607516191508865, −3.30862581511211526808665064607, −3.19602483629408867223959706149, −2.60682931451682078377866780442, −2.27462517979186196670706891729, −0.899719447387035424950288718152, −0.56218169268051479198333510255,
0.56218169268051479198333510255, 0.899719447387035424950288718152, 2.27462517979186196670706891729, 2.60682931451682078377866780442, 3.19602483629408867223959706149, 3.30862581511211526808665064607, 3.71259277831989607516191508865, 4.32884756750620426229543927381, 4.37328754425237463248456331961, 5.18832742479788427609666569673, 5.58971411688820460309784520000, 6.02359901027962498270680217781, 6.28366043460594445186935420235, 6.58207218974627329522711917673, 6.76507104220192588980680516194, 7.23033937902870630758178903027, 7.86335525327547243594087859369, 8.069197698724880143985706539150, 8.635439420297348243659689296025, 8.812982482223015601622394317940
