| L(s) = 1 | − 2-s + 2.12·3-s + 4-s − 0.444·5-s − 2.12·6-s + 0.813·7-s − 8-s + 1.53·9-s + 0.444·10-s + 2.12·12-s + 0.531·13-s − 0.813·14-s − 0.946·15-s + 16-s − 5.88·17-s − 1.53·18-s − 19-s − 0.444·20-s + 1.73·21-s + 6.69·23-s − 2.12·24-s − 4.80·25-s − 0.531·26-s − 3.12·27-s + 0.813·28-s − 4.16·29-s + 0.946·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.22·3-s + 0.5·4-s − 0.198·5-s − 0.869·6-s + 0.307·7-s − 0.353·8-s + 0.510·9-s + 0.140·10-s + 0.614·12-s + 0.147·13-s − 0.217·14-s − 0.244·15-s + 0.250·16-s − 1.42·17-s − 0.360·18-s − 0.229·19-s − 0.0994·20-s + 0.378·21-s + 1.39·23-s − 0.434·24-s − 0.960·25-s − 0.104·26-s − 0.601·27-s + 0.153·28-s − 0.773·29-s + 0.172·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 + 0.444T + 5T^{2} \) |
| 7 | \( 1 - 0.813T + 7T^{2} \) |
| 13 | \( 1 - 0.531T + 13T^{2} \) |
| 17 | \( 1 + 5.88T + 17T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 - 2.37T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 - 3.37T + 59T^{2} \) |
| 61 | \( 1 + 4.76T + 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.219239979769288439043308646945, −7.33710859378639568736608906390, −6.92098131975446002089067485096, −5.88641030321470845932358796596, −4.91337325768479948353145537192, −3.92168203015765784775500243998, −3.21721555650249476053745248713, −2.28292448104231810278364958309, −1.62911797152089149980957944918, 0,
1.62911797152089149980957944918, 2.28292448104231810278364958309, 3.21721555650249476053745248713, 3.92168203015765784775500243998, 4.91337325768479948353145537192, 5.88641030321470845932358796596, 6.92098131975446002089067485096, 7.33710859378639568736608906390, 8.219239979769288439043308646945
