| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4.24·7-s − 8-s + 9-s − 10-s − 1.91·11-s − 12-s − 4.24·14-s − 15-s + 16-s − 3.33·17-s − 18-s + 4.85·19-s + 20-s − 4.24·21-s + 1.91·22-s − 0.445·23-s + 24-s + 25-s − 27-s + 4.24·28-s − 8.56·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.60·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.576·11-s − 0.288·12-s − 1.13·14-s − 0.258·15-s + 0.250·16-s − 0.808·17-s − 0.235·18-s + 1.11·19-s + 0.223·20-s − 0.926·21-s + 0.407·22-s − 0.0927·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.802·28-s − 1.59·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 + 0.445T + 23T^{2} \) |
| 29 | \( 1 + 8.56T + 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 3.24T + 41T^{2} \) |
| 43 | \( 1 + 1.97T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 1.13T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 2.14T + 71T^{2} \) |
| 73 | \( 1 + 5.15T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 - 1.25T + 89T^{2} \) |
| 97 | \( 1 - 5.01T + 97T^{2} \) |
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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
−7.72931920499020830492441181188, −7.43478198496802426582989574338, −6.49931163443829656394400916073, −5.56309444106375095118007637567, −5.16487859340103535737479613759, −4.34585808968207905504375677561, −3.13113147031746869658640837313, −1.91062247819935596059076956954, −1.48552666373572429276451379960, 0,
1.48552666373572429276451379960, 1.91062247819935596059076956954, 3.13113147031746869658640837313, 4.34585808968207905504375677561, 5.16487859340103535737479613759, 5.56309444106375095118007637567, 6.49931163443829656394400916073, 7.43478198496802426582989574338, 7.72931920499020830492441181188
