| L(s) = 1 | + 7-s + 2·19-s − 2·25-s + 4·31-s + 2·37-s − 2·103-s + 4·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 7-s + 2·19-s − 2·25-s + 4·31-s + 2·37-s − 2·103-s + 4·109-s − 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.849846394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.849846394\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.982122033918700693180425142677, −8.763316056971802860603705060973, −8.059212534706248217699421073676, −8.000082823805304229918377380967, −7.71157923380136627476009284378, −7.48770033423719360694296207017, −6.66826377480308245791367754243, −6.54877473425715344413765516201, −6.00340544153279344395039359802, −5.67464738173274534164290899973, −5.27440097455641778916618109346, −4.78225903292187444384168970905, −4.35283196893819456604085755785, −4.26844645739728218305114578073, −3.42563022764564594435307600199, −3.11301935564982350208081780204, −2.47455616865384170663430799329, −2.19330522920098157853743573032, −1.17049495654480585624863669411, −1.09046079678066803011179607190,
1.09046079678066803011179607190, 1.17049495654480585624863669411, 2.19330522920098157853743573032, 2.47455616865384170663430799329, 3.11301935564982350208081780204, 3.42563022764564594435307600199, 4.26844645739728218305114578073, 4.35283196893819456604085755785, 4.78225903292187444384168970905, 5.27440097455641778916618109346, 5.67464738173274534164290899973, 6.00340544153279344395039359802, 6.54877473425715344413765516201, 6.66826377480308245791367754243, 7.48770033423719360694296207017, 7.71157923380136627476009284378, 8.000082823805304229918377380967, 8.059212534706248217699421073676, 8.763316056971802860603705060973, 8.982122033918700693180425142677
