| L(s) = 1 | − 3.03·3-s + 2.46·7-s + 6.19·9-s − 0.728·11-s + 6.23·13-s − 0.563·17-s + 19-s − 7.49·21-s + 4.63·23-s − 9.70·27-s + 10.2·29-s − 6.06·31-s + 2.21·33-s + 5.72·37-s − 18.9·39-s + 4.79·41-s + 8.06·43-s + 8.12·47-s − 0.900·49-s + 1.70·51-s + 1.53·53-s − 3.03·57-s + 5.76·59-s + 10.9·61-s + 15.3·63-s + 12.9·67-s − 14.0·69-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 0.933·7-s + 2.06·9-s − 0.219·11-s + 1.72·13-s − 0.136·17-s + 0.229·19-s − 1.63·21-s + 0.966·23-s − 1.86·27-s + 1.89·29-s − 1.08·31-s + 0.384·33-s + 0.941·37-s − 3.02·39-s + 0.748·41-s + 1.23·43-s + 1.18·47-s − 0.128·49-s + 0.239·51-s + 0.210·53-s − 0.401·57-s + 0.750·59-s + 1.40·61-s + 1.92·63-s + 1.58·67-s − 1.69·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.592482773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.592482773\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 0.728T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.563T + 17T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 6.06T + 31T^{2} \) |
| 37 | \( 1 - 5.72T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.85T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.82142753175158629334238182109, −6.89367407945250005510228424173, −6.49021737792074664408740277859, −5.52884321930316538402834562022, −5.40480110732223830788625411585, −4.40094898341595333935509734015, −3.93192501361106391519041576464, −2.57893078085054921112968065139, −1.27018085369507755373299784058, −0.845526121389797306699601912610,
0.845526121389797306699601912610, 1.27018085369507755373299784058, 2.57893078085054921112968065139, 3.93192501361106391519041576464, 4.40094898341595333935509734015, 5.40480110732223830788625411585, 5.52884321930316538402834562022, 6.49021737792074664408740277859, 6.89367407945250005510228424173, 7.82142753175158629334238182109
