| L(s) = 1 | + 2·3-s + 3·9-s − 10·19-s − 2·25-s + 4·27-s + 10·31-s − 22·37-s + 12·47-s + 24·53-s − 20·57-s − 24·59-s − 4·75-s + 5·81-s − 36·83-s + 20·93-s + 10·103-s − 14·109-s − 44·111-s − 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 24·141-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.29·19-s − 2/5·25-s + 0.769·27-s + 1.79·31-s − 3.61·37-s + 1.75·47-s + 3.29·53-s − 2.64·57-s − 3.12·59-s − 0.461·75-s + 5/9·81-s − 3.95·83-s + 2.07·93-s + 0.985·103-s − 1.34·109-s − 4.17·111-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.02·141-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.697536638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697536638\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.409863477005114765813697953850, −8.724419783202658386347118158521, −8.406825439633831400881201267445, −8.346557778780206596675586920066, −7.62622813778368307672790550775, −7.24366023348999556481093641612, −7.03054961388024306225890718685, −6.43282934919576666577401721914, −6.29677724126691030585079720232, −5.47021868472603074523878989202, −5.39929973568992063165783240205, −4.52389517164719368798028541267, −4.26276336308682154602837989785, −4.00913351942227560404143369319, −3.43425875021722192515375802399, −2.75765159939395673957920250623, −2.63509226060092959808889592459, −1.78546889058288277760838198860, −1.65609198074412514788105944751, −0.49293210020831007669943317233,
0.49293210020831007669943317233, 1.65609198074412514788105944751, 1.78546889058288277760838198860, 2.63509226060092959808889592459, 2.75765159939395673957920250623, 3.43425875021722192515375802399, 4.00913351942227560404143369319, 4.26276336308682154602837989785, 4.52389517164719368798028541267, 5.39929973568992063165783240205, 5.47021868472603074523878989202, 6.29677724126691030585079720232, 6.43282934919576666577401721914, 7.03054961388024306225890718685, 7.24366023348999556481093641612, 7.62622813778368307672790550775, 8.346557778780206596675586920066, 8.406825439633831400881201267445, 8.724419783202658386347118158521, 9.409863477005114765813697953850
