| L(s) = 1 | − 0.820·2-s + 2.32·3-s − 1.32·4-s + 5-s − 1.90·6-s − 0.561·7-s + 2.72·8-s + 2.41·9-s − 0.820·10-s − 0.111·11-s − 3.08·12-s − 6.89·13-s + 0.460·14-s + 2.32·15-s + 0.415·16-s − 3.81·17-s − 1.98·18-s − 1.32·20-s − 1.30·21-s + 0.0911·22-s − 4.04·23-s + 6.35·24-s + 25-s + 5.65·26-s − 1.35·27-s + 0.745·28-s − 9.29·29-s + ⋯ |
| L(s) = 1 | − 0.580·2-s + 1.34·3-s − 0.663·4-s + 0.447·5-s − 0.779·6-s − 0.212·7-s + 0.964·8-s + 0.805·9-s − 0.259·10-s − 0.0334·11-s − 0.891·12-s − 1.91·13-s + 0.123·14-s + 0.600·15-s + 0.103·16-s − 0.924·17-s − 0.467·18-s − 0.296·20-s − 0.285·21-s + 0.0194·22-s − 0.842·23-s + 1.29·24-s + 0.200·25-s + 1.10·26-s − 0.261·27-s + 0.140·28-s − 1.72·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.820T + 2T^{2} \) |
| 3 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 + 0.111T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 - 5.74T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.64T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 + 8.16T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−8.997441775051851374421843156593, −8.189230799826952465039978745125, −7.62295373388420760953512645529, −6.84554802373447528433998608220, −5.51919845452052674602712926015, −4.61611401453096682719702119475, −3.76742027376767595715760253449, −2.58998853805246055365973835122, −1.88494929449626927902308617843, 0,
1.88494929449626927902308617843, 2.58998853805246055365973835122, 3.76742027376767595715760253449, 4.61611401453096682719702119475, 5.51919845452052674602712926015, 6.84554802373447528433998608220, 7.62295373388420760953512645529, 8.189230799826952465039978745125, 8.997441775051851374421843156593
