| L(s) = 1 | − 4.90·2-s + 0.777·3-s + 16.1·4-s − 9.19·5-s − 3.81·6-s − 39.7·8-s − 26.3·9-s + 45.1·10-s + 22.4·11-s + 12.5·12-s + 62.0·13-s − 7.15·15-s + 66.4·16-s + 72.1·17-s + 129.·18-s + 25.0·19-s − 148.·20-s − 110.·22-s + 103.·23-s − 30.9·24-s − 40.4·25-s − 304.·26-s − 41.5·27-s − 234.·29-s + 35.1·30-s − 198.·31-s − 8.17·32-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 0.149·3-s + 2.01·4-s − 0.822·5-s − 0.259·6-s − 1.75·8-s − 0.977·9-s + 1.42·10-s + 0.614·11-s + 0.301·12-s + 1.32·13-s − 0.123·15-s + 1.03·16-s + 1.02·17-s + 1.69·18-s + 0.302·19-s − 1.65·20-s − 1.06·22-s + 0.935·23-s − 0.263·24-s − 0.323·25-s − 2.29·26-s − 0.296·27-s − 1.50·29-s + 0.213·30-s − 1.15·31-s − 0.0451·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 + 4.90T + 8T^{2} \) |
| 3 | \( 1 - 0.777T + 27T^{2} \) |
| 5 | \( 1 + 9.19T + 125T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 111.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 499.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 562.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 760.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 975.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 698.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 451.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 390.T + 9.12e5T^{2} \) |
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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
−8.487215562108334590762178484144, −7.65218899324940553793650283656, −7.18987530338448402264622565424, −6.17151983094887486846541122730, −5.40444136842827528059477395603, −3.77128422246405960976960388949, −3.24232469971577685779499603281, −1.89305390626045416237344965601, −0.976858488604196520562315548070, 0,
0.976858488604196520562315548070, 1.89305390626045416237344965601, 3.24232469971577685779499603281, 3.77128422246405960976960388949, 5.40444136842827528059477395603, 6.17151983094887486846541122730, 7.18987530338448402264622565424, 7.65218899324940553793650283656, 8.487215562108334590762178484144
