| L(s) = 1 | + 3-s + 4-s − 5-s − 7-s + 9-s + 12-s − 15-s + 16-s − 19-s − 20-s − 21-s − 23-s + 27-s − 28-s − 29-s + 35-s + 36-s − 37-s + 2·41-s − 45-s − 47-s + 48-s + 2·53-s − 57-s − 60-s − 61-s − 63-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 5-s − 7-s + 9-s + 12-s − 15-s + 16-s − 19-s − 20-s − 21-s − 23-s + 27-s − 28-s − 29-s + 35-s + 36-s − 37-s + 2·41-s − 45-s − 47-s + 48-s + 2·53-s − 57-s − 60-s − 61-s − 63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.093491175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093491175\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.47525477433014055819585806148, −10.50165252681353449018808221758, −9.670101870211429215377047795001, −8.564670901488282840563327165507, −7.70311784044715877802272763779, −7.00777087059962764495445353100, −6.01138996447162082756729418329, −4.11736566165569695423514503991, −3.34740423745780496534310739551, −2.16992743686203330013514209476,
2.16992743686203330013514209476, 3.34740423745780496534310739551, 4.11736566165569695423514503991, 6.01138996447162082756729418329, 7.00777087059962764495445353100, 7.70311784044715877802272763779, 8.564670901488282840563327165507, 9.670101870211429215377047795001, 10.50165252681353449018808221758, 11.47525477433014055819585806148
