| L(s) = 1 | + 8·11-s + 12·19-s + 4·31-s + 4·41-s − 49-s − 8·59-s − 4·61-s + 16·71-s + 16·79-s − 20·89-s − 36·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 2.75·19-s + 0.718·31-s + 0.624·41-s − 1/7·49-s − 1.04·59-s − 0.512·61-s + 1.89·71-s + 1.80·79-s − 2.11·89-s − 3.58·101-s + 3.44·109-s + 2.36·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.227912494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.227912494\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.029108341433804675935561384812, −8.021279672209943821296496575223, −7.36217632658600584537799457631, −7.19873817642127874573907845842, −6.73897017268950462542234465879, −6.61013293141771302146961983835, −5.97913968347512812799097645355, −5.90739157938273370534202987837, −5.32345802526354629006121808286, −5.06611641492879727186260982347, −4.47395119402303431507444268712, −4.29429827690586609545079306104, −3.63328223819437140553514499652, −3.60172147668940021473344557986, −2.95338828867454867458710314230, −2.77480262112386879330596720038, −1.80914884569144606538662082038, −1.62610686909669391694051832092, −0.926987279583274509990761737795, −0.75185787420926044424284759151,
0.75185787420926044424284759151, 0.926987279583274509990761737795, 1.62610686909669391694051832092, 1.80914884569144606538662082038, 2.77480262112386879330596720038, 2.95338828867454867458710314230, 3.60172147668940021473344557986, 3.63328223819437140553514499652, 4.29429827690586609545079306104, 4.47395119402303431507444268712, 5.06611641492879727186260982347, 5.32345802526354629006121808286, 5.90739157938273370534202987837, 5.97913968347512812799097645355, 6.61013293141771302146961983835, 6.73897017268950462542234465879, 7.19873817642127874573907845842, 7.36217632658600584537799457631, 8.021279672209943821296496575223, 8.029108341433804675935561384812
