| L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s + 4·13-s + 16-s − 8·19-s + 2·21-s + 25-s + 27-s + 2·28-s − 8·31-s + 36-s + 4·37-s + 4·39-s + 16·43-s + 48-s + 3·49-s + 4·52-s − 8·57-s + 4·61-s + 2·63-s + 64-s + 16·67-s + 28·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 1.43·31-s + 1/6·36-s + 0.657·37-s + 0.640·39-s + 2.43·43-s + 0.144·48-s + 3/7·49-s + 0.554·52-s − 1.05·57-s + 0.512·61-s + 0.251·63-s + 1/8·64-s + 1.95·67-s + 3.27·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.518951744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.518951744\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.268646864684040562273313820184, −8.800046649726438641176234759863, −8.319627980737908137853866947983, −8.134234480169490737550982107259, −7.27965583167809929942184453108, −7.16665739633458789232295323445, −6.15797813375789816669713446161, −6.14620005069006563789988203374, −5.29787033315398374470104513854, −4.64849576847169223614699526425, −3.88231405811014601850145796336, −3.71822253537069739639390686069, −2.51414806834535899531438368517, −2.16105217917237072156284139202, −1.18226526293597327102151626443,
1.18226526293597327102151626443, 2.16105217917237072156284139202, 2.51414806834535899531438368517, 3.71822253537069739639390686069, 3.88231405811014601850145796336, 4.64849576847169223614699526425, 5.29787033315398374470104513854, 6.14620005069006563789988203374, 6.15797813375789816669713446161, 7.16665739633458789232295323445, 7.27965583167809929942184453108, 8.134234480169490737550982107259, 8.319627980737908137853866947983, 8.800046649726438641176234759863, 9.268646864684040562273313820184
