| L(s) = 1 | + 2·3-s + 3·9-s + 6·13-s − 4·17-s − 16·23-s + 6·25-s + 4·27-s + 4·29-s + 12·39-s − 16·43-s − 2·49-s − 8·51-s + 12·53-s + 4·61-s − 32·69-s + 12·75-s + 5·81-s + 8·87-s − 20·101-s − 8·103-s + 40·107-s − 20·113-s + 18·117-s − 14·121-s + 127-s − 32·129-s + 131-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.66·13-s − 0.970·17-s − 3.33·23-s + 6/5·25-s + 0.769·27-s + 0.742·29-s + 1.92·39-s − 2.43·43-s − 2/7·49-s − 1.12·51-s + 1.64·53-s + 0.512·61-s − 3.85·69-s + 1.38·75-s + 5/9·81-s + 0.857·87-s − 1.99·101-s − 0.788·103-s + 3.86·107-s − 1.88·113-s + 1.66·117-s − 1.27·121-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.765233348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765233348\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−13.26657220693466661731948892325, −12.94932386452613100433413645942, −12.08276686225715194715834263929, −11.88375396217184662205761068204, −11.12128195663621032232420364802, −10.60968435213668987297176670796, −9.984272626526803271025881911865, −9.793052053774398617031321730766, −8.733357697731066924305144083051, −8.634512573800313818344453651741, −8.218875798418116076630631770765, −7.61638043898165400830288508122, −6.69800751741339245219009105952, −6.43439919378819890018806124998, −5.67311703943898644966786049773, −4.69084294487850614818309091038, −3.96615499698305412496572344175, −3.54941629286267712884198762723, −2.51497422957877804194212538733, −1.65252021016815052804364632200,
1.65252021016815052804364632200, 2.51497422957877804194212538733, 3.54941629286267712884198762723, 3.96615499698305412496572344175, 4.69084294487850614818309091038, 5.67311703943898644966786049773, 6.43439919378819890018806124998, 6.69800751741339245219009105952, 7.61638043898165400830288508122, 8.218875798418116076630631770765, 8.634512573800313818344453651741, 8.733357697731066924305144083051, 9.793052053774398617031321730766, 9.984272626526803271025881911865, 10.60968435213668987297176670796, 11.12128195663621032232420364802, 11.88375396217184662205761068204, 12.08276686225715194715834263929, 12.94932386452613100433413645942, 13.26657220693466661731948892325
