| L(s) = 1 | − 4-s − 9-s + 6·11-s + 16-s − 2·19-s − 6·29-s − 14·31-s + 36-s − 12·41-s − 6·44-s + 10·49-s − 12·59-s − 2·61-s − 64-s + 2·76-s + 26·79-s + 81-s + 30·89-s − 6·99-s + 12·101-s − 4·109-s + 6·116-s + 5·121-s + 14·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.80·11-s + 1/4·16-s − 0.458·19-s − 1.11·29-s − 2.51·31-s + 1/6·36-s − 1.87·41-s − 0.904·44-s + 10/7·49-s − 1.56·59-s − 0.256·61-s − 1/8·64-s + 0.229·76-s + 2.92·79-s + 1/9·81-s + 3.17·89-s − 0.603·99-s + 1.19·101-s − 0.383·109-s + 0.557·116-s + 5/11·121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.434076097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.434076097\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.960817171424932230850591668881, −8.925817151104560719677690922187, −8.203689491785621569435049598151, −7.85867432769430594325865061508, −7.49794897295285178928755009453, −7.06360070851082435006210612849, −6.69293565861219500582445708262, −6.28568428421864848302951154897, −5.97489717608363614423259902517, −5.51763079022218854793002672080, −4.95256193733036729163702073838, −4.88125390593818221945071210739, −3.93595653881679154302446511767, −3.91615768826487210993454157480, −3.52307290822718916342623165467, −3.06242873533281056358605333291, −2.05829890358335335103380825973, −1.92977440294093514106609649805, −1.24117565967959199251996470866, −0.40478713874505623865881763221,
0.40478713874505623865881763221, 1.24117565967959199251996470866, 1.92977440294093514106609649805, 2.05829890358335335103380825973, 3.06242873533281056358605333291, 3.52307290822718916342623165467, 3.91615768826487210993454157480, 3.93595653881679154302446511767, 4.88125390593818221945071210739, 4.95256193733036729163702073838, 5.51763079022218854793002672080, 5.97489717608363614423259902517, 6.28568428421864848302951154897, 6.69293565861219500582445708262, 7.06360070851082435006210612849, 7.49794897295285178928755009453, 7.85867432769430594325865061508, 8.203689491785621569435049598151, 8.925817151104560719677690922187, 8.960817171424932230850591668881
