| L(s) = 1 | − 4-s − 9-s + 16-s − 25-s − 4·31-s + 36-s + 2·49-s − 64-s + 4·79-s + 81-s + 100-s − 2·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s − 9-s + 16-s − 25-s − 4·31-s + 36-s + 2·49-s − 64-s + 4·79-s + 81-s + 100-s − 2·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2956273915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2956273915\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.21271855810921919646769460812, −13.33009494655484601976149952531, −13.19667196832582720061135768379, −12.39747538269979838031491661615, −12.09747784861115319834900421262, −11.41393770282462266589397115446, −10.68268464629724600508738886889, −10.61437846076570269693066417743, −9.388643639095434605845356081616, −9.386450812199118293810282841415, −8.817852444608721014831468079651, −8.108885684803283632899512093538, −7.63338202461196921745610868863, −6.96322967351268565631502757622, −5.91048570812207665106258742322, −5.56402930318338092618140678491, −4.97657542880787386510676331574, −3.86980282416878206705418894327, −3.52472146273302481464633973046, −2.15313362465143136876203290842,
2.15313362465143136876203290842, 3.52472146273302481464633973046, 3.86980282416878206705418894327, 4.97657542880787386510676331574, 5.56402930318338092618140678491, 5.91048570812207665106258742322, 6.96322967351268565631502757622, 7.63338202461196921745610868863, 8.108885684803283632899512093538, 8.817852444608721014831468079651, 9.386450812199118293810282841415, 9.388643639095434605845356081616, 10.61437846076570269693066417743, 10.68268464629724600508738886889, 11.41393770282462266589397115446, 12.09747784861115319834900421262, 12.39747538269979838031491661615, 13.19667196832582720061135768379, 13.33009494655484601976149952531, 14.21271855810921919646769460812
