| L(s) = 1 | − 2-s + 3·5-s + 8-s − 3·10-s + 3·11-s − 4·13-s − 16-s + 6·17-s + 2·19-s − 3·22-s + 6·23-s + 5·25-s + 4·26-s + 18·29-s − 7·31-s − 6·34-s + 10·37-s − 2·38-s + 3·40-s − 8·43-s − 6·46-s + 12·47-s − 5·50-s + 3·53-s + 9·55-s − 18·58-s − 3·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.904·11-s − 1.10·13-s − 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.639·22-s + 1.25·23-s + 25-s + 0.784·26-s + 3.34·29-s − 1.25·31-s − 1.02·34-s + 1.64·37-s − 0.324·38-s + 0.474·40-s − 1.21·43-s − 0.884·46-s + 1.75·47-s − 0.707·50-s + 0.412·53-s + 1.21·55-s − 2.36·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.173708473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173708473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 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
−10.06502239234059846024934120753, −9.914596816739796655964406422655, −9.621097700032801685287275279727, −9.167106426148207105421252136291, −8.658877678390949271719062878088, −8.563048714366870042694140334932, −7.64066317557623715206834195122, −7.56376278038186079532245472403, −6.83721184947006595217123994578, −6.67861577314183203538960806192, −5.87141457385585522235718061394, −5.76352586846298603187543162755, −4.95322795892157370763371436472, −4.79081928730827179568301296600, −4.12030687197011669242125277882, −3.21730886961047019684902764107, −2.83615087649566079448549012224, −2.22817976872039375862690908606, −1.27769304516479923611064999072, −0.971510353541376040801985068443,
0.971510353541376040801985068443, 1.27769304516479923611064999072, 2.22817976872039375862690908606, 2.83615087649566079448549012224, 3.21730886961047019684902764107, 4.12030687197011669242125277882, 4.79081928730827179568301296600, 4.95322795892157370763371436472, 5.76352586846298603187543162755, 5.87141457385585522235718061394, 6.67861577314183203538960806192, 6.83721184947006595217123994578, 7.56376278038186079532245472403, 7.64066317557623715206834195122, 8.563048714366870042694140334932, 8.658877678390949271719062878088, 9.167106426148207105421252136291, 9.621097700032801685287275279727, 9.914596816739796655964406422655, 10.06502239234059846024934120753
