| L(s) = 1 | + 2·3-s − 3·9-s + 6·17-s − 18·23-s − 25-s − 14·27-s − 18·29-s + 2·43-s + 13·49-s + 12·51-s + 12·53-s − 2·61-s − 36·69-s − 2·75-s + 16·79-s − 4·81-s − 36·87-s − 30·101-s − 16·103-s − 6·107-s − 30·113-s + 13·121-s + 127-s + 4·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s + 1.45·17-s − 3.75·23-s − 1/5·25-s − 2.69·27-s − 3.34·29-s + 0.304·43-s + 13/7·49-s + 1.68·51-s + 1.64·53-s − 0.256·61-s − 4.33·69-s − 0.230·75-s + 1.80·79-s − 4/9·81-s − 3.85·87-s − 2.98·101-s − 1.57·103-s − 0.580·107-s − 2.82·113-s + 1.18·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6383858021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6383858021\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.747822336928920243967286541707, −8.240828408226520802480695744429, −8.119835155430687636170451690378, −7.79671815510750244378478594292, −7.55935750465578414101877219972, −7.15942214953401994021429701168, −6.50650617511480785901082878294, −6.00701714687442570923136873491, −5.68715628429157400917095491878, −5.48150112663575078687124018658, −5.35379742121607885638979907732, −4.20605763088023793278058537453, −3.94036531501304038414612871862, −3.71554123870115520137958058362, −3.43727263550882673983140698158, −2.57347257545187325681429332499, −2.41956503605614468970415125290, −1.97565966585360190061995054872, −1.39743196763718808093819162818, −0.20794397042005058094806288444,
0.20794397042005058094806288444, 1.39743196763718808093819162818, 1.97565966585360190061995054872, 2.41956503605614468970415125290, 2.57347257545187325681429332499, 3.43727263550882673983140698158, 3.71554123870115520137958058362, 3.94036531501304038414612871862, 4.20605763088023793278058537453, 5.35379742121607885638979907732, 5.48150112663575078687124018658, 5.68715628429157400917095491878, 6.00701714687442570923136873491, 6.50650617511480785901082878294, 7.15942214953401994021429701168, 7.55935750465578414101877219972, 7.79671815510750244378478594292, 8.119835155430687636170451690378, 8.240828408226520802480695744429, 8.747822336928920243967286541707
