| L(s) = 1 | − 1.96·2-s + 3-s + 1.88·4-s + 4.31·5-s − 1.96·6-s − 7-s + 0.234·8-s + 9-s − 8.50·10-s − 2.03·11-s + 1.88·12-s + 3.99·13-s + 1.96·14-s + 4.31·15-s − 4.22·16-s + 5.63·17-s − 1.96·18-s + 6.75·19-s + 8.12·20-s − 21-s + 4.00·22-s + 1.90·23-s + 0.234·24-s + 13.6·25-s − 7.86·26-s + 27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.577·3-s + 0.940·4-s + 1.93·5-s − 0.804·6-s − 0.377·7-s + 0.0830·8-s + 0.333·9-s − 2.68·10-s − 0.612·11-s + 0.542·12-s + 1.10·13-s + 0.526·14-s + 1.11·15-s − 1.05·16-s + 1.36·17-s − 0.464·18-s + 1.54·19-s + 1.81·20-s − 0.218·21-s + 0.853·22-s + 0.396·23-s + 0.0479·24-s + 2.72·25-s − 1.54·26-s + 0.192·27-s − 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.905762655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905762655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 - 4.31T + 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 - 6.75T + 19T^{2} \) |
| 23 | \( 1 - 1.90T + 23T^{2} \) |
| 29 | \( 1 + 3.06T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 0.254T + 41T^{2} \) |
| 43 | \( 1 - 6.07T + 43T^{2} \) |
| 47 | \( 1 + 9.20T + 47T^{2} \) |
| 53 | \( 1 - 9.42T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 + 7.37T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.849629718840505304904074471730, −7.63509447865212353528839982190, −7.45525476079773883347581175553, −6.23573186051875100270726771164, −5.75152627379929223017200013929, −4.91406772463321667437796323137, −3.40964678671905145761606445166, −2.67010328069507393172415505462, −1.63601334429134223175272736220, −1.07405121225096352784948846772,
1.07405121225096352784948846772, 1.63601334429134223175272736220, 2.67010328069507393172415505462, 3.40964678671905145761606445166, 4.91406772463321667437796323137, 5.75152627379929223017200013929, 6.23573186051875100270726771164, 7.45525476079773883347581175553, 7.63509447865212353528839982190, 8.849629718840505304904074471730
