| L(s) = 1 | − 0.583·2-s + 3.17·3-s − 1.65·4-s + 1.76·5-s − 1.85·6-s + 2.13·8-s + 7.09·9-s − 1.03·10-s − 2.47·11-s − 5.27·12-s − 4.76·13-s + 5.61·15-s + 2.07·16-s + 2.27·17-s − 4.13·18-s + 5.68·19-s − 2.93·20-s + 1.44·22-s + 1.88·23-s + 6.78·24-s − 1.88·25-s + 2.78·26-s + 12.9·27-s − 10.0·29-s − 3.27·30-s − 4.26·31-s − 5.47·32-s + ⋯ |
| L(s) = 1 | − 0.412·2-s + 1.83·3-s − 0.829·4-s + 0.789·5-s − 0.756·6-s + 0.754·8-s + 2.36·9-s − 0.325·10-s − 0.746·11-s − 1.52·12-s − 1.32·13-s + 1.44·15-s + 0.518·16-s + 0.552·17-s − 0.975·18-s + 1.30·19-s − 0.655·20-s + 0.308·22-s + 0.392·23-s + 1.38·24-s − 0.376·25-s + 0.545·26-s + 2.50·27-s − 1.86·29-s − 0.597·30-s − 0.765·31-s − 0.968·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.105343426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.105343426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.583T + 2T^{2} \) |
| 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 23 | \( 1 - 1.88T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.115T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 8.97T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.77T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
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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
−7.934732790922474012380596289590, −7.57231070183778962229841846784, −7.26885823208167831022480904727, −5.66851298871165690783217417718, −5.20789103389435332708459526001, −4.20303645516664963758561976933, −3.55436740909482885233691075652, −2.58131592459792640396850136066, −2.08318204149755615080839398895, −0.923061332191924758043801083397,
0.923061332191924758043801083397, 2.08318204149755615080839398895, 2.58131592459792640396850136066, 3.55436740909482885233691075652, 4.20303645516664963758561976933, 5.20789103389435332708459526001, 5.66851298871165690783217417718, 7.26885823208167831022480904727, 7.57231070183778962229841846784, 7.934732790922474012380596289590
