| L(s) = 1 | + 4-s + 2·9-s − 4·11-s − 4·19-s + 2·36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s − 4·76-s + 81-s − 2·89-s − 8·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 8·171-s + ⋯ |
| L(s) = 1 | + 4-s + 2·9-s − 4·11-s − 4·19-s + 2·36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s − 4·76-s + 81-s − 2·89-s − 8·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 8·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0004180032235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0004180032235\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.17603316675996591401192436771, −5.96894356491835295057725185524, −5.91875364966140385447326713004, −5.79992265497494044270326951877, −5.32957707757763564411021223001, −5.06065578028761016041746096033, −4.93279233357205246477165754355, −4.79794174952914353150191007824, −4.61715981751012187409223125149, −4.60351665941689216392655804704, −4.11791924272882553828010741323, −3.98056424565927286662877702585, −3.95294462717315068304958301786, −3.56484028768031980897174616828, −3.11053773880054685238066262751, −2.96105634288664350026406552648, −2.63055915712198760707282549783, −2.60176423012445083647290627784, −2.55032613297760839768431632473, −2.07291369267733819593939412870, −1.81120807287227144280810705385, −1.68754997756578918708510113751, −1.57257215932757804336470389349, −0.930489133371709867450600470670, −0.00672197456157567302788981606,
0.00672197456157567302788981606, 0.930489133371709867450600470670, 1.57257215932757804336470389349, 1.68754997756578918708510113751, 1.81120807287227144280810705385, 2.07291369267733819593939412870, 2.55032613297760839768431632473, 2.60176423012445083647290627784, 2.63055915712198760707282549783, 2.96105634288664350026406552648, 3.11053773880054685238066262751, 3.56484028768031980897174616828, 3.95294462717315068304958301786, 3.98056424565927286662877702585, 4.11791924272882553828010741323, 4.60351665941689216392655804704, 4.61715981751012187409223125149, 4.79794174952914353150191007824, 4.93279233357205246477165754355, 5.06065578028761016041746096033, 5.32957707757763564411021223001, 5.79992265497494044270326951877, 5.91875364966140385447326713004, 5.96894356491835295057725185524, 6.17603316675996591401192436771
