| L(s) = 1 | + 16·17-s − 4·25-s + 8·41-s − 12·49-s − 32·73-s − 18·81-s + 32·89-s + 48·97-s − 40·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.88·17-s − 4/5·25-s + 1.24·41-s − 1.71·49-s − 3.74·73-s − 2·81-s + 3.39·89-s + 4.87·97-s − 3.76·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.790664272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.790664272\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69266512566471474131032569388, −7.65509637645303748716351251470, −7.59322659337975999846162825759, −7.46516106364732171144236711857, −6.81013914606852552902760015025, −6.57626597887461464462409569998, −6.54267158937272220441311735576, −5.95208375245605713397282951722, −5.88746771670892961778490739202, −5.64221698087336796276120606526, −5.58078468071887640583841488005, −5.12168398279132397412047199929, −4.94282607564432262407122132913, −4.45366493274182597385844681408, −4.41789960736634217717900753342, −3.99876985566059403257786034123, −3.45040604903541744310470265052, −3.41340310559163132884020181631, −3.24905262027254111206400767098, −2.82726879506261276505943020536, −2.49530337705237636829552183797, −1.82352945209908855423593239363, −1.63499018798053959517818857357, −1.12086998888940462161330765088, −0.62198359824054923700143727844,
0.62198359824054923700143727844, 1.12086998888940462161330765088, 1.63499018798053959517818857357, 1.82352945209908855423593239363, 2.49530337705237636829552183797, 2.82726879506261276505943020536, 3.24905262027254111206400767098, 3.41340310559163132884020181631, 3.45040604903541744310470265052, 3.99876985566059403257786034123, 4.41789960736634217717900753342, 4.45366493274182597385844681408, 4.94282607564432262407122132913, 5.12168398279132397412047199929, 5.58078468071887640583841488005, 5.64221698087336796276120606526, 5.88746771670892961778490739202, 5.95208375245605713397282951722, 6.54267158937272220441311735576, 6.57626597887461464462409569998, 6.81013914606852552902760015025, 7.46516106364732171144236711857, 7.59322659337975999846162825759, 7.65509637645303748716351251470, 7.69266512566471474131032569388
