L(s) = 1 | + 2-s − 2.41·3-s + 4-s + 5-s − 2.41·6-s − 7-s + 8-s + 2.82·9-s + 10-s − 2.41·12-s + 2.76·13-s − 14-s − 2.41·15-s + 16-s + 4.44·17-s + 2.82·18-s − 2.62·19-s + 20-s + 2.41·21-s + 1.54·23-s − 2.41·24-s + 25-s + 2.76·26-s + 0.414·27-s − 28-s − 9.32·29-s − 2.41·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.447·5-s − 0.985·6-s − 0.377·7-s + 0.353·8-s + 0.942·9-s + 0.316·10-s − 0.696·12-s + 0.767·13-s − 0.267·14-s − 0.623·15-s + 0.250·16-s + 1.07·17-s + 0.666·18-s − 0.601·19-s + 0.223·20-s + 0.526·21-s + 0.322·23-s − 0.492·24-s + 0.200·25-s + 0.542·26-s + 0.0797·27-s − 0.188·28-s − 1.73·29-s − 0.440·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 13 | \( 1 - 2.76T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.62T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.39T + 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 9.77T + 53T^{2} \) |
| 59 | \( 1 - 0.510T + 59T^{2} \) |
| 61 | \( 1 + 2.63T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 3.41T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 4.27T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 2.35T + 97T^{2} \) |
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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
−7.09213138524350748609996180708, −6.51177565473416724959530321983, −5.96524976082356566283673664467, −5.33151121290417656266965437426, −5.04185239532623729312200667249, −3.80430401437602265860659752393, −3.43145572254642024806820909105, −2.11178366081477015658998927270, −1.26307135938770776399751758257, 0,
1.26307135938770776399751758257, 2.11178366081477015658998927270, 3.43145572254642024806820909105, 3.80430401437602265860659752393, 5.04185239532623729312200667249, 5.33151121290417656266965437426, 5.96524976082356566283673664467, 6.51177565473416724959530321983, 7.09213138524350748609996180708