| L(s) = 1 | + 1.70·2-s − 3·3-s − 5.10·4-s − 5.10·6-s − 22.2·7-s − 22.2·8-s + 9·9-s − 1.79·11-s + 15.3·12-s − 58.2·13-s − 37.7·14-s + 2.89·16-s − 18.9·17-s + 15.3·18-s + 104.·19-s + 66.6·21-s − 3.04·22-s + 49.6·23-s + 66.8·24-s − 99.0·26-s − 27·27-s + 113.·28-s − 293.·29-s + 64.4·31-s + 183.·32-s + 5.37·33-s − 32.3·34-s + ⋯ |
| L(s) = 1 | + 0.601·2-s − 0.577·3-s − 0.638·4-s − 0.347·6-s − 1.19·7-s − 0.985·8-s + 0.333·9-s − 0.0490·11-s + 0.368·12-s − 1.24·13-s − 0.721·14-s + 0.0452·16-s − 0.270·17-s + 0.200·18-s + 1.26·19-s + 0.692·21-s − 0.0295·22-s + 0.449·23-s + 0.568·24-s − 0.747·26-s − 0.192·27-s + 0.765·28-s − 1.87·29-s + 0.373·31-s + 1.01·32-s + 0.0283·33-s − 0.162·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.70T + 8T^{2} \) |
| 7 | \( 1 + 22.2T + 343T^{2} \) |
| 11 | \( 1 + 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 19.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 173.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 384.T + 9.12e5T^{2} \) |
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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
−13.27157851785774737093994881337, −12.63258333376567957900508221671, −11.59815975981730049363946513900, −9.951828766338302418789239651135, −9.256654948930752207899454969180, −7.34095213332534839345331384556, −5.96673464896432176929292865722, −4.82682767214552604076343977457, −3.26221395701708635052479657648, 0,
3.26221395701708635052479657648, 4.82682767214552604076343977457, 5.96673464896432176929292865722, 7.34095213332534839345331384556, 9.256654948930752207899454969180, 9.951828766338302418789239651135, 11.59815975981730049363946513900, 12.63258333376567957900508221671, 13.27157851785774737093994881337
