| L(s) = 1 | + 7-s + 2·13-s − 3·19-s + 25-s + 3·31-s + 37-s − 2·61-s − 3·67-s − 73-s + 3·79-s + 2·91-s − 4·97-s − 3·103-s − 109-s − 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | + 7-s + 2·13-s − 3·19-s + 25-s + 3·31-s + 37-s − 2·61-s − 3·67-s − 73-s + 3·79-s + 2·91-s − 4·97-s − 3·103-s − 109-s − 121-s + 127-s + 131-s − 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.187743052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187743052\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
| 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
−10.42155080946776623291331768250, −10.24510717110406885069276913112, −9.261681444660892454132230085584, −9.207418794365086457917037546001, −8.426468370268175024764815754304, −8.405080512342872457790042388779, −8.112762697373264439384570786860, −7.65259079006108557485197483987, −6.71531333641253673567420261415, −6.60628435969078377498052152443, −6.08562677816785593985410502332, −5.94398593579786385967362859058, −5.00343132322739890688605824133, −4.64540335931995716799197097869, −4.08133127446727830685973476746, −4.05197065307616256362195188110, −2.82279822552738431504329786803, −2.72902753207584850539372492885, −1.64777075275712270920995495399, −1.24909024554417592940450729484,
1.24909024554417592940450729484, 1.64777075275712270920995495399, 2.72902753207584850539372492885, 2.82279822552738431504329786803, 4.05197065307616256362195188110, 4.08133127446727830685973476746, 4.64540335931995716799197097869, 5.00343132322739890688605824133, 5.94398593579786385967362859058, 6.08562677816785593985410502332, 6.60628435969078377498052152443, 6.71531333641253673567420261415, 7.65259079006108557485197483987, 8.112762697373264439384570786860, 8.405080512342872457790042388779, 8.426468370268175024764815754304, 9.207418794365086457917037546001, 9.261681444660892454132230085584, 10.24510717110406885069276913112, 10.42155080946776623291331768250
