| L(s) = 1 | + 2.38·2-s + 3-s + 3.67·4-s + 3.76·5-s + 2.38·6-s − 1.09·7-s + 3.99·8-s + 9-s + 8.95·10-s − 11-s + 3.67·12-s − 1.78·13-s − 2.60·14-s + 3.76·15-s + 2.16·16-s − 3.04·17-s + 2.38·18-s − 3.42·19-s + 13.8·20-s − 1.09·21-s − 2.38·22-s + 6.13·23-s + 3.99·24-s + 9.14·25-s − 4.24·26-s + 27-s − 4.01·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.577·3-s + 1.83·4-s + 1.68·5-s + 0.972·6-s − 0.412·7-s + 1.41·8-s + 0.333·9-s + 2.83·10-s − 0.301·11-s + 1.06·12-s − 0.494·13-s − 0.694·14-s + 0.970·15-s + 0.540·16-s − 0.738·17-s + 0.561·18-s − 0.786·19-s + 3.09·20-s − 0.238·21-s − 0.507·22-s + 1.27·23-s + 0.815·24-s + 1.82·25-s − 0.832·26-s + 0.192·27-s − 0.758·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.985341045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.985341045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 + 3.04T + 17T^{2} \) |
| 19 | \( 1 + 3.42T + 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 0.248T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 0.212T + 73T^{2} \) |
| 79 | \( 1 + 5.71T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 8.41T + 89T^{2} \) |
| 97 | \( 1 + 6.22T + 97T^{2} \) |
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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
−9.244328426562862958334471576236, −8.438949733463557380950913242110, −7.08637393035786622116992837200, −6.51219571026960354746372924402, −5.93288012780333717316444255926, −4.96637777995698590333925655841, −4.48450160836576536569172422931, −3.08727175255515598739115595071, −2.62838770271108860047486644580, −1.72873311016211870714252304271,
1.72873311016211870714252304271, 2.62838770271108860047486644580, 3.08727175255515598739115595071, 4.48450160836576536569172422931, 4.96637777995698590333925655841, 5.93288012780333717316444255926, 6.51219571026960354746372924402, 7.08637393035786622116992837200, 8.438949733463557380950913242110, 9.244328426562862958334471576236
