| L(s) = 1 | + 2-s + 0.779·3-s + 4-s − 5-s + 0.779·6-s − 2.22·7-s + 8-s − 2.39·9-s − 10-s + 2.97·11-s + 0.779·12-s − 13-s − 2.22·14-s − 0.779·15-s + 16-s − 0.203·17-s − 2.39·18-s + 3.28·19-s − 20-s − 1.73·21-s + 2.97·22-s − 3.80·23-s + 0.779·24-s + 25-s − 26-s − 4.20·27-s − 2.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.450·3-s + 0.5·4-s − 0.447·5-s + 0.318·6-s − 0.839·7-s + 0.353·8-s − 0.797·9-s − 0.316·10-s + 0.898·11-s + 0.225·12-s − 0.277·13-s − 0.593·14-s − 0.201·15-s + 0.250·16-s − 0.0492·17-s − 0.563·18-s + 0.753·19-s − 0.223·20-s − 0.377·21-s + 0.635·22-s − 0.793·23-s + 0.159·24-s + 0.200·25-s − 0.196·26-s − 0.809·27-s − 0.419·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 0.779T + 3T^{2} \) |
| 7 | \( 1 + 2.22T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 17 | \( 1 + 0.203T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 6.54T + 29T^{2} \) |
| 37 | \( 1 + 5.84T + 37T^{2} \) |
| 41 | \( 1 + 8.74T + 41T^{2} \) |
| 43 | \( 1 - 1.81T + 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 2.59T + 71T^{2} \) |
| 73 | \( 1 - 1.18T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 14.0T + 83T^{2} \) |
| 89 | \( 1 - 0.495T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
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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
−7.992162139058389036960898482607, −7.27457646126686933949795806785, −6.54873717918851008596508238250, −5.85324142552406664019084870712, −5.08853936267471805392195311140, −4.00319539667116745460095259567, −3.47422783410218608540524557055, −2.79422178037703051817540858948, −1.67125960773575846618900742645, 0,
1.67125960773575846618900742645, 2.79422178037703051817540858948, 3.47422783410218608540524557055, 4.00319539667116745460095259567, 5.08853936267471805392195311140, 5.85324142552406664019084870712, 6.54873717918851008596508238250, 7.27457646126686933949795806785, 7.992162139058389036960898482607
