| L(s) = 1 | − 2·2-s + 4·4-s + 3.44·5-s − 8·8-s − 6.89·10-s + 36.1·11-s − 10.2·13-s + 16·16-s − 118.·17-s + 38.6·19-s + 13.7·20-s − 72.2·22-s − 36.2·23-s − 113.·25-s + 20.4·26-s − 12.1·29-s + 145.·31-s − 32·32-s + 236.·34-s − 1.37·37-s − 77.3·38-s − 27.5·40-s − 168·41-s + 299.·43-s + 144.·44-s + 72.4·46-s − 502.·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.308·5-s − 0.353·8-s − 0.217·10-s + 0.990·11-s − 0.218·13-s + 0.250·16-s − 1.69·17-s + 0.466·19-s + 0.154·20-s − 0.700·22-s − 0.328·23-s − 0.904·25-s + 0.154·26-s − 0.0776·29-s + 0.842·31-s − 0.176·32-s + 1.19·34-s − 0.00609·37-s − 0.330·38-s − 0.108·40-s − 0.639·41-s + 1.06·43-s + 0.495·44-s + 0.232·46-s − 1.56·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.44T + 125T^{2} \) |
| 11 | \( 1 - 36.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 36.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 12.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.37T + 5.06e4T^{2} \) |
| 41 | \( 1 + 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 502.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 625.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 42.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 763.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 579.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 942.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 474.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 821.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.10e3T + 9.12e5T^{2} \) |
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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
−9.387467350183254222444125160846, −8.612704906059426280862084524540, −7.73482935857194203677128015887, −6.69347659385950267891146737612, −6.18476559577194290316695853626, −4.87574529762245979549440910788, −3.79430313647371614291145859694, −2.45462485663241713426549120986, −1.44501379208625896681188716243, 0,
1.44501379208625896681188716243, 2.45462485663241713426549120986, 3.79430313647371614291145859694, 4.87574529762245979549440910788, 6.18476559577194290316695853626, 6.69347659385950267891146737612, 7.73482935857194203677128015887, 8.612704906059426280862084524540, 9.387467350183254222444125160846
