| L(s) = 1 | + 2.47·2-s + 3·3-s − 1.89·4-s + 4.05·5-s + 7.41·6-s − 24.4·8-s + 9·9-s + 10.0·10-s − 11·11-s − 5.67·12-s + 64.2·13-s + 12.1·15-s − 45.2·16-s + 26.0·17-s + 22.2·18-s − 85.8·19-s − 7.68·20-s − 27.1·22-s − 106.·23-s − 73.3·24-s − 108.·25-s + 158.·26-s + 27·27-s − 269.·29-s + 30.0·30-s − 2.76·31-s + 83.6·32-s + ⋯ |
| L(s) = 1 | + 0.873·2-s + 0.577·3-s − 0.236·4-s + 0.363·5-s + 0.504·6-s − 1.08·8-s + 0.333·9-s + 0.317·10-s − 0.301·11-s − 0.136·12-s + 1.37·13-s + 0.209·15-s − 0.707·16-s + 0.370·17-s + 0.291·18-s − 1.03·19-s − 0.0859·20-s − 0.263·22-s − 0.966·23-s − 0.623·24-s − 0.868·25-s + 1.19·26-s + 0.192·27-s − 1.72·29-s + 0.183·30-s − 0.0160·31-s + 0.462·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.47T + 8T^{2} \) |
| 5 | \( 1 - 4.05T + 125T^{2} \) |
| 13 | \( 1 - 64.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.76T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 229.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 81.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 252.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 477.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 433.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 14.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 902.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 631.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 707.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 646.T + 9.12e5T^{2} \) |
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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
−8.504028938325846316336114543851, −8.094323894257939109602805398034, −6.83088581862152517419148056120, −5.92203878595972236475740231990, −5.44652396032895202055778660329, −4.10796958743258667833621293890, −3.80704444289471282941098294054, −2.66710445742998833663601424101, −1.60801097823514929117880016144, 0,
1.60801097823514929117880016144, 2.66710445742998833663601424101, 3.80704444289471282941098294054, 4.10796958743258667833621293890, 5.44652396032895202055778660329, 5.92203878595972236475740231990, 6.83088581862152517419148056120, 8.094323894257939109602805398034, 8.504028938325846316336114543851
