| L(s) = 1 | + 3-s − 6·5-s − 2·7-s + 9-s − 6·15-s − 2·17-s − 2·21-s + 18·25-s + 27-s + 12·35-s + 4·37-s + 14·41-s + 20·43-s − 6·45-s − 4·47-s − 3·49-s − 2·51-s − 2·63-s − 4·67-s + 18·75-s − 4·79-s + 81-s − 12·83-s + 12·85-s − 10·89-s − 14·101-s + 12·105-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.68·5-s − 0.755·7-s + 1/3·9-s − 1.54·15-s − 0.485·17-s − 0.436·21-s + 18/5·25-s + 0.192·27-s + 2.02·35-s + 0.657·37-s + 2.18·41-s + 3.04·43-s − 0.894·45-s − 0.583·47-s − 3/7·49-s − 0.280·51-s − 0.251·63-s − 0.488·67-s + 2.07·75-s − 0.450·79-s + 1/9·81-s − 1.31·83-s + 1.30·85-s − 1.05·89-s − 1.39·101-s + 1.17·105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 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
−8.526886250018319278022984247612, −7.916444938192842770152464531629, −7.66296019082410579715472600226, −7.28977396910275299746230088556, −6.96875433625664370593302474467, −6.16932034978540659008329630263, −5.82315016489928418237652191479, −4.81857255802540354856377091023, −4.28849821783009308772395142489, −4.06829240979909057979509560486, −3.61245895574183336724282338933, −2.91023400853945447052213713658, −2.50851125103879833016047936182, −1.00386786647596745975951806846, 0,
1.00386786647596745975951806846, 2.50851125103879833016047936182, 2.91023400853945447052213713658, 3.61245895574183336724282338933, 4.06829240979909057979509560486, 4.28849821783009308772395142489, 4.81857255802540354856377091023, 5.82315016489928418237652191479, 6.16932034978540659008329630263, 6.96875433625664370593302474467, 7.28977396910275299746230088556, 7.66296019082410579715472600226, 7.916444938192842770152464531629, 8.526886250018319278022984247612
