| L(s) = 1 | + 1.29·2-s − 3.36·3-s − 6.32·4-s + 1.78·5-s − 4.34·6-s − 27.5·7-s − 18.5·8-s − 15.7·9-s + 2.30·10-s + 9.45·11-s + 21.2·12-s + 38.4·13-s − 35.6·14-s − 5.98·15-s + 26.6·16-s + 54.0·17-s − 20.3·18-s + 37.3·19-s − 11.2·20-s + 92.6·21-s + 12.2·22-s − 100.·23-s + 62.2·24-s − 121.·25-s + 49.7·26-s + 143.·27-s + 174.·28-s + ⋯ |
| L(s) = 1 | + 0.457·2-s − 0.646·3-s − 0.790·4-s + 0.159·5-s − 0.295·6-s − 1.48·7-s − 0.819·8-s − 0.581·9-s + 0.0729·10-s + 0.259·11-s + 0.511·12-s + 0.820·13-s − 0.681·14-s − 0.103·15-s + 0.415·16-s + 0.771·17-s − 0.266·18-s + 0.450·19-s − 0.125·20-s + 0.962·21-s + 0.118·22-s − 0.915·23-s + 0.529·24-s − 0.974·25-s + 0.375·26-s + 1.02·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.29T + 8T^{2} \) |
| 3 | \( 1 + 3.36T + 27T^{2} \) |
| 5 | \( 1 - 1.78T + 125T^{2} \) |
| 7 | \( 1 + 27.5T + 343T^{2} \) |
| 11 | \( 1 - 9.45T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 234.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 325.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 148.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 735.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 275.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 30.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 27.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 66.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 144.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 769.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
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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
−8.673389213073413328841288439466, −7.68531227034512394330319354682, −6.52604737972396470513358450742, −5.78607611759116594581513606574, −5.64406477233472084603088224442, −4.25021572807759312746108034957, −3.57621553424830592450677373702, −2.74192738426718430821686019131, −0.925377485845133948859926572905, 0,
0.925377485845133948859926572905, 2.74192738426718430821686019131, 3.57621553424830592450677373702, 4.25021572807759312746108034957, 5.64406477233472084603088224442, 5.78607611759116594581513606574, 6.52604737972396470513358450742, 7.68531227034512394330319354682, 8.673389213073413328841288439466
