| L(s) = 1 | + 2-s + 2.44·3-s + 4-s − 4.14·5-s + 2.44·6-s + 8-s + 2.96·9-s − 4.14·10-s − 4.70·11-s + 2.44·12-s + 1.91·13-s − 10.1·15-s + 16-s − 2.76·17-s + 2.96·18-s − 4.04·19-s − 4.14·20-s − 4.70·22-s − 2.17·23-s + 2.44·24-s + 12.1·25-s + 1.91·26-s − 0.0841·27-s − 29-s − 10.1·30-s + 0.969·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s − 1.85·5-s + 0.997·6-s + 0.353·8-s + 0.988·9-s − 1.31·10-s − 1.41·11-s + 0.705·12-s + 0.529·13-s − 2.61·15-s + 0.250·16-s − 0.671·17-s + 0.698·18-s − 0.926·19-s − 0.926·20-s − 1.00·22-s − 0.453·23-s + 0.498·24-s + 2.43·25-s + 0.374·26-s − 0.0161·27-s − 0.185·29-s − 1.84·30-s + 0.174·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 31 | \( 1 - 0.969T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 + 3.41T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 6.24T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 - 6.06T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.407932515611875582067630547205, −7.71838180988731143123506778602, −7.20848040035940420312451754245, −6.19765393047081426532635150267, −4.88098556482957925673773739166, −4.30117814300836855188890856755, −3.47659756898704189107202459999, −2.99206435617086103305650882671, −1.98374511470141636639797059169, 0,
1.98374511470141636639797059169, 2.99206435617086103305650882671, 3.47659756898704189107202459999, 4.30117814300836855188890856755, 4.88098556482957925673773739166, 6.19765393047081426532635150267, 7.20848040035940420312451754245, 7.71838180988731143123506778602, 8.407932515611875582067630547205
