| L(s) = 1 | + 2·3-s − 6·5-s − 3·9-s − 11-s − 12·15-s − 6·23-s + 17·25-s − 14·27-s + 10·31-s − 2·33-s − 2·37-s + 18·45-s − 10·49-s − 12·53-s + 6·55-s + 6·59-s − 2·67-s − 12·69-s + 30·71-s + 34·75-s − 4·81-s − 18·89-s + 20·93-s − 14·97-s + 3·99-s + 16·103-s − 4·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.68·5-s − 9-s − 0.301·11-s − 3.09·15-s − 1.25·23-s + 17/5·25-s − 2.69·27-s + 1.79·31-s − 0.348·33-s − 0.328·37-s + 2.68·45-s − 1.42·49-s − 1.64·53-s + 0.809·55-s + 0.781·59-s − 0.244·67-s − 1.44·69-s + 3.56·71-s + 3.92·75-s − 4/9·81-s − 1.90·89-s + 2.07·93-s − 1.42·97-s + 0.301·99-s + 1.57·103-s − 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.82109708424629217833914870894, −9.813753565180213813561671136388, −9.431312841027853518235538734747, −8.453584718504860075735545131686, −8.204138017445950847285230309260, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −6.82620843631036387433675479847, −6.07532856213092694592866222076, −5.11227132195889762323685903220, −4.33075028432463196258949288256, −3.68366753852883183133176875421, −3.27381985666804460223319703873, −2.47303941619863806873155456193, 0,
2.47303941619863806873155456193, 3.27381985666804460223319703873, 3.68366753852883183133176875421, 4.33075028432463196258949288256, 5.11227132195889762323685903220, 6.07532856213092694592866222076, 6.82620843631036387433675479847, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.204138017445950847285230309260, 8.453584718504860075735545131686, 9.431312841027853518235538734747, 9.813753565180213813561671136388, 10.82109708424629217833914870894
