| L(s) = 1 | + 2-s + 0.870·3-s + 4-s + 1.72·5-s + 0.870·6-s + 2.12·7-s + 8-s − 2.24·9-s + 1.72·10-s + 5.37·11-s + 0.870·12-s − 0.704·13-s + 2.12·14-s + 1.49·15-s + 16-s + 17-s − 2.24·18-s − 0.612·19-s + 1.72·20-s + 1.84·21-s + 5.37·22-s + 1.69·23-s + 0.870·24-s − 2.03·25-s − 0.704·26-s − 4.56·27-s + 2.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.502·3-s + 0.5·4-s + 0.770·5-s + 0.355·6-s + 0.802·7-s + 0.353·8-s − 0.747·9-s + 0.544·10-s + 1.62·11-s + 0.251·12-s − 0.195·13-s + 0.567·14-s + 0.386·15-s + 0.250·16-s + 0.242·17-s − 0.528·18-s − 0.140·19-s + 0.385·20-s + 0.403·21-s + 1.14·22-s + 0.352·23-s + 0.177·24-s − 0.406·25-s − 0.138·26-s − 0.877·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.210513523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.210513523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.870T + 3T^{2} \) |
| 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 + 0.704T + 13T^{2} \) |
| 19 | \( 1 + 0.612T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 61 | \( 1 - 5.29T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 - 0.0795T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.931276818346863432612413831120, −8.603179270545841987175989651124, −7.47562520967991251738485392677, −6.69573229553620848996319575304, −5.85325064475926056900951531947, −5.19738197999166667761005100035, −4.17913239761658361483601417892, −3.35063684853523036756171093787, −2.26595763216785624253984523881, −1.42748433652646299444938209826,
1.42748433652646299444938209826, 2.26595763216785624253984523881, 3.35063684853523036756171093787, 4.17913239761658361483601417892, 5.19738197999166667761005100035, 5.85325064475926056900951531947, 6.69573229553620848996319575304, 7.47562520967991251738485392677, 8.603179270545841987175989651124, 8.931276818346863432612413831120
