| L(s) = 1 | + 2·3-s − 4-s + 3·9-s − 2·12-s − 6·13-s + 16-s − 2·17-s + 6·23-s + 9·25-s + 4·27-s + 18·29-s − 3·36-s − 12·39-s + 14·43-s + 2·48-s − 4·51-s + 6·52-s − 20·53-s − 22·61-s − 64-s + 2·68-s + 12·69-s + 18·75-s − 24·79-s + 5·81-s + 36·87-s − 6·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 1.66·13-s + 1/4·16-s − 0.485·17-s + 1.25·23-s + 9/5·25-s + 0.769·27-s + 3.34·29-s − 1/2·36-s − 1.92·39-s + 2.13·43-s + 0.288·48-s − 0.560·51-s + 0.832·52-s − 2.74·53-s − 2.81·61-s − 1/8·64-s + 0.242·68-s + 1.44·69-s + 2.07·75-s − 2.70·79-s + 5/9·81-s + 3.85·87-s − 0.625·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.669082913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.669082913\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−8.562691603506545187861969681773, −8.487904997384668041396432615618, −8.040799761435534634006907861439, −7.58639437470812525863821908552, −7.20180303009482840078556347604, −6.99611118554134547191010237314, −6.67084508926905075459820608989, −6.00505162139473931424560633859, −5.86026904655368839996563599443, −4.90287299670456806488712143333, −4.75196329634317413850083645205, −4.48283135021068261159319022096, −4.46848613547646762787045116358, −3.35631710749628737351974869296, −3.13971455647621508470567506090, −2.67851837991112713246543037524, −2.62496960532589786557000241104, −1.74282238501729628405472542383, −1.16383456892299597767969462023, −0.56782214118402675752941473852,
0.56782214118402675752941473852, 1.16383456892299597767969462023, 1.74282238501729628405472542383, 2.62496960532589786557000241104, 2.67851837991112713246543037524, 3.13971455647621508470567506090, 3.35631710749628737351974869296, 4.46848613547646762787045116358, 4.48283135021068261159319022096, 4.75196329634317413850083645205, 4.90287299670456806488712143333, 5.86026904655368839996563599443, 6.00505162139473931424560633859, 6.67084508926905075459820608989, 6.99611118554134547191010237314, 7.20180303009482840078556347604, 7.58639437470812525863821908552, 8.040799761435534634006907861439, 8.487904997384668041396432615618, 8.562691603506545187861969681773
