| L(s) = 1 | − 9-s − 8·11-s + 4·19-s + 12·29-s − 20·31-s + 16·41-s + 10·49-s + 16·59-s − 28·61-s + 32·71-s + 32·79-s + 81-s + 8·89-s + 8·99-s + 20·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 2.41·11-s + 0.917·19-s + 2.22·29-s − 3.59·31-s + 2.49·41-s + 10/7·49-s + 2.08·59-s − 3.58·61-s + 3.79·71-s + 3.60·79-s + 1/9·81-s + 0.847·89-s + 0.804·99-s + 1.99·101-s + 0.383·109-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.002651920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.002651920\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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
−8.569060842297272741035026183049, −8.198937019865660966338759335951, −7.84485410849878976438568082881, −7.64230268158386858570712331887, −7.29960857867754195702741577097, −6.99033619779418207256481860607, −6.33704629284539194715826325860, −6.03737650264777878047540587594, −5.46959890701307364531605799044, −5.44481559349346003748052322742, −4.87545886170949793420720091993, −4.76078851271347277465880954350, −3.99279750073826273698741766308, −3.53476094087088779480373907759, −3.23688193190466955137982623707, −2.61347705753606061775168101310, −2.30954123229753188494097466945, −1.97610104160655761645026826401, −0.881909319301179468339753103370, −0.51504408080624484691603387277,
0.51504408080624484691603387277, 0.881909319301179468339753103370, 1.97610104160655761645026826401, 2.30954123229753188494097466945, 2.61347705753606061775168101310, 3.23688193190466955137982623707, 3.53476094087088779480373907759, 3.99279750073826273698741766308, 4.76078851271347277465880954350, 4.87545886170949793420720091993, 5.44481559349346003748052322742, 5.46959890701307364531605799044, 6.03737650264777878047540587594, 6.33704629284539194715826325860, 6.99033619779418207256481860607, 7.29960857867754195702741577097, 7.64230268158386858570712331887, 7.84485410849878976438568082881, 8.198937019865660966338759335951, 8.569060842297272741035026183049
