| L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s − 12·13-s + 8·15-s + 11·25-s − 4·27-s + 16·31-s − 4·37-s + 24·39-s − 16·41-s + 12·43-s − 12·45-s − 49-s − 12·53-s + 48·65-s + 4·67-s + 16·71-s − 22·75-s − 20·79-s + 5·81-s − 8·83-s − 32·93-s + 24·107-s + 8·111-s − 36·117-s + 22·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s − 3.32·13-s + 2.06·15-s + 11/5·25-s − 0.769·27-s + 2.87·31-s − 0.657·37-s + 3.84·39-s − 2.49·41-s + 1.82·43-s − 1.78·45-s − 1/7·49-s − 1.64·53-s + 5.95·65-s + 0.488·67-s + 1.89·71-s − 2.54·75-s − 2.25·79-s + 5/9·81-s − 0.878·83-s − 3.31·93-s + 2.32·107-s + 0.759·111-s − 3.32·117-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3990053560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3990053560\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.682811177184510242258009697753, −8.330930547511987971438085556120, −7.80531120093154186454057323951, −7.77841685468538486035892709426, −7.23329015721359118440342809506, −6.96567477981761374923079124435, −6.76772507938308963759707337596, −6.28699624342385626985278406297, −5.73419146841517276423694697831, −5.16868697802840566543630991686, −4.82702048847931642479359662314, −4.73957362977125156160147748847, −4.37000814100310825777023089899, −3.89895878746919370239977757652, −3.19149967919829601241504471599, −2.89560237545425143389815825411, −2.38518871903012722702557574717, −1.69732792631369941281081976029, −0.74833576553058149841603470573, −0.30728108029786993346128404027,
0.30728108029786993346128404027, 0.74833576553058149841603470573, 1.69732792631369941281081976029, 2.38518871903012722702557574717, 2.89560237545425143389815825411, 3.19149967919829601241504471599, 3.89895878746919370239977757652, 4.37000814100310825777023089899, 4.73957362977125156160147748847, 4.82702048847931642479359662314, 5.16868697802840566543630991686, 5.73419146841517276423694697831, 6.28699624342385626985278406297, 6.76772507938308963759707337596, 6.96567477981761374923079124435, 7.23329015721359118440342809506, 7.77841685468538486035892709426, 7.80531120093154186454057323951, 8.330930547511987971438085556120, 8.682811177184510242258009697753
