| L(s) = 1 | − 2-s + 3.40·3-s + 4-s + 1.22·5-s − 3.40·6-s − 7-s − 8-s + 8.57·9-s − 1.22·10-s + 4.94·11-s + 3.40·12-s + 0.454·13-s + 14-s + 4.17·15-s + 16-s + 7.25·17-s − 8.57·18-s + 1.22·20-s − 3.40·21-s − 4.94·22-s − 1.54·23-s − 3.40·24-s − 3.49·25-s − 0.454·26-s + 18.9·27-s − 28-s + 4.03·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.96·3-s + 0.5·4-s + 0.548·5-s − 1.38·6-s − 0.377·7-s − 0.353·8-s + 2.85·9-s − 0.388·10-s + 1.49·11-s + 0.982·12-s + 0.125·13-s + 0.267·14-s + 1.07·15-s + 0.250·16-s + 1.76·17-s − 2.02·18-s + 0.274·20-s − 0.742·21-s − 1.05·22-s − 0.322·23-s − 0.694·24-s − 0.698·25-s − 0.0890·26-s + 3.65·27-s − 0.188·28-s + 0.748·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.958303961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.958303961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 - 0.454T + 13T^{2} \) |
| 17 | \( 1 - 7.25T + 17T^{2} \) |
| 23 | \( 1 + 1.54T + 23T^{2} \) |
| 29 | \( 1 - 4.03T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 + 8.03T + 37T^{2} \) |
| 41 | \( 1 - 2.77T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 + 0.597T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 - 0.143T + 61T^{2} \) |
| 67 | \( 1 - 1.25T + 67T^{2} \) |
| 71 | \( 1 - 0.948T + 71T^{2} \) |
| 73 | \( 1 + 7.89T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−8.473251687529174822544393777815, −7.61999741486185590915675929587, −7.11829218665174295411469283501, −6.39220138371895678815047168560, −5.42256527849905467539032842402, −4.01654029170240912114765982187, −3.57390365221725030212230399090, −2.80840502971182451281457896406, −1.78839269224868925123001060336, −1.28171662422903756973197579049,
1.28171662422903756973197579049, 1.78839269224868925123001060336, 2.80840502971182451281457896406, 3.57390365221725030212230399090, 4.01654029170240912114765982187, 5.42256527849905467539032842402, 6.39220138371895678815047168560, 7.11829218665174295411469283501, 7.61999741486185590915675929587, 8.473251687529174822544393777815
