| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·9-s − 4·11-s + 5·16-s − 8·18-s − 8·22-s − 8·23-s − 2·25-s + 4·29-s + 6·32-s − 12·36-s + 20·37-s + 4·43-s − 12·44-s − 16·46-s − 4·50-s − 4·53-s + 8·58-s + 7·64-s + 24·67-s − 24·71-s − 16·72-s + 40·74-s − 8·79-s + 7·81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s − 1.20·11-s + 5/4·16-s − 1.88·18-s − 1.70·22-s − 1.66·23-s − 2/5·25-s + 0.742·29-s + 1.06·32-s − 2·36-s + 3.28·37-s + 0.609·43-s − 1.80·44-s − 2.35·46-s − 0.565·50-s − 0.549·53-s + 1.05·58-s + 7/8·64-s + 2.93·67-s − 2.84·71-s − 1.88·72-s + 4.64·74-s − 0.900·79-s + 7/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.896963851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.896963851\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.08889201258632171177809522471, −13.75897892764682158104944478298, −12.99181020642601878454218038210, −12.96065310228585205937755138639, −11.91815093620013056118481872743, −11.88110889862418557601494564222, −11.08749216798728719416053267125, −10.79739703594736369657619807538, −9.981854834006967455041537620725, −9.508258444468192616988098371622, −8.371218444153554891246935980326, −8.041979252561494532787930581368, −7.48845653750310791794370134824, −6.48255113005534387242997547576, −5.86451827160416506222883333068, −5.60287084867711068938249624170, −4.66415541315340900968485969565, −4.02076711545202908644935151541, −2.90852999167922493658293827195, −2.41717302449154006608420706146,
2.41717302449154006608420706146, 2.90852999167922493658293827195, 4.02076711545202908644935151541, 4.66415541315340900968485969565, 5.60287084867711068938249624170, 5.86451827160416506222883333068, 6.48255113005534387242997547576, 7.48845653750310791794370134824, 8.041979252561494532787930581368, 8.371218444153554891246935980326, 9.508258444468192616988098371622, 9.981854834006967455041537620725, 10.79739703594736369657619807538, 11.08749216798728719416053267125, 11.88110889862418557601494564222, 11.91815093620013056118481872743, 12.96065310228585205937755138639, 12.99181020642601878454218038210, 13.75897892764682158104944478298, 14.08889201258632171177809522471
