| L(s) = 1 | − 0.928·2-s + 4.15·3-s − 7.13·4-s − 7.22·5-s − 3.86·6-s − 33.4·7-s + 14.0·8-s − 9.70·9-s + 6.70·10-s + 21.8·11-s − 29.6·12-s − 37.7·13-s + 31.1·14-s − 30.0·15-s + 44.0·16-s + 87.4·17-s + 9.01·18-s + 115.·19-s + 51.5·20-s − 139.·21-s − 20.2·22-s − 3.50·23-s + 58.4·24-s − 72.8·25-s + 35.1·26-s − 152.·27-s + 239.·28-s + ⋯ |
| L(s) = 1 | − 0.328·2-s + 0.800·3-s − 0.892·4-s − 0.645·5-s − 0.262·6-s − 1.80·7-s + 0.621·8-s − 0.359·9-s + 0.212·10-s + 0.597·11-s − 0.714·12-s − 0.806·13-s + 0.593·14-s − 0.516·15-s + 0.688·16-s + 1.24·17-s + 0.118·18-s + 1.39·19-s + 0.576·20-s − 1.44·21-s − 0.196·22-s − 0.0317·23-s + 0.497·24-s − 0.582·25-s + 0.264·26-s − 1.08·27-s + 1.61·28-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 | \( 1 + 0.928T + 8T^{2} \) |
| 3 | \( 1 - 4.15T + 27T^{2} \) |
| 5 | \( 1 + 7.22T + 125T^{2} \) |
| 7 | \( 1 + 33.4T + 343T^{2} \) |
| 11 | \( 1 - 21.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 37.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.50T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 446.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 526.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 575.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 112.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 6.38T + 3.00e5T^{2} \) |
| 71 | \( 1 + 570.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 424.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 395.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 472.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 836.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 640.T + 9.12e5T^{2} \) |
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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
−8.607169083646458367712958488620, −7.78082528313627556433322874195, −7.23703058771225503047946935928, −6.10870867757101615737138028488, −5.23992987465138179787212690795, −4.00264860097396840796037091447, −3.41182125702676138740521387882, −2.75515027412120210450905481481, −0.959372897747489400776356758491, 0,
0.959372897747489400776356758491, 2.75515027412120210450905481481, 3.41182125702676138740521387882, 4.00264860097396840796037091447, 5.23992987465138179787212690795, 6.10870867757101615737138028488, 7.23703058771225503047946935928, 7.78082528313627556433322874195, 8.607169083646458367712958488620
