| L(s) = 1 | − 6·7-s + 5·9-s + 6·17-s + 18·23-s + 10·25-s − 12·31-s + 12·41-s + 13·49-s − 30·63-s + 22·73-s + 24·79-s + 16·81-s − 16·97-s − 12·103-s − 24·113-s − 36·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s − 108·161-s + 163-s + ⋯ |
| L(s) = 1 | − 2.26·7-s + 5/3·9-s + 1.45·17-s + 3.75·23-s + 2·25-s − 2.15·31-s + 1.87·41-s + 13/7·49-s − 3.77·63-s + 2.57·73-s + 2.70·79-s + 16/9·81-s − 1.62·97-s − 1.18·103-s − 2.25·113-s − 3.30·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s − 8.51·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.433738716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433738716\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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
−9.661165550558986193511889780429, −9.452082351369450189442281696885, −9.338297278609692927807365341382, −9.064455935931297151545476565667, −8.312700185860270809450382307879, −7.77307587699736122115391753624, −7.16052590256910063539794378756, −7.08070584991519485848520445672, −6.69205073326629599870617058586, −6.50250727412608230979435762145, −5.55270838919019823822788404583, −5.43690883218433792305781071075, −4.79260310657974610597233093823, −4.35336587019880043428892678064, −3.46764598695459551704989318974, −3.39567998387535460812683799868, −3.01000527337709967171374865417, −2.24773209460051432385704753352, −1.09513157184347448438907192422, −0.859629915268150574693685825194,
0.859629915268150574693685825194, 1.09513157184347448438907192422, 2.24773209460051432385704753352, 3.01000527337709967171374865417, 3.39567998387535460812683799868, 3.46764598695459551704989318974, 4.35336587019880043428892678064, 4.79260310657974610597233093823, 5.43690883218433792305781071075, 5.55270838919019823822788404583, 6.50250727412608230979435762145, 6.69205073326629599870617058586, 7.08070584991519485848520445672, 7.16052590256910063539794378756, 7.77307587699736122115391753624, 8.312700185860270809450382307879, 9.064455935931297151545476565667, 9.338297278609692927807365341382, 9.452082351369450189442281696885, 9.661165550558986193511889780429
