| L(s) = 1 | − 2-s − 3-s + 4-s + 2.27·5-s + 6-s − 8-s + 9-s − 2.27·10-s − 11-s − 12-s + 4.63·13-s − 2.27·15-s + 16-s − 0.554·17-s − 18-s − 4.07·19-s + 2.27·20-s + 22-s − 6.93·23-s + 24-s + 0.171·25-s − 4.63·26-s − 27-s − 2.82·29-s + 2.27·30-s − 7.68·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.01·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.719·10-s − 0.301·11-s − 0.288·12-s + 1.28·13-s − 0.587·15-s + 0.250·16-s − 0.134·17-s − 0.235·18-s − 0.935·19-s + 0.508·20-s + 0.213·22-s − 1.44·23-s + 0.204·24-s + 0.0343·25-s − 0.908·26-s − 0.192·27-s − 0.525·29-s + 0.415·30-s − 1.38·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.27T + 5T^{2} \) |
| 13 | \( 1 - 4.63T + 13T^{2} \) |
| 17 | \( 1 + 0.554T + 17T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + 6.93T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 0.554T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.440T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−8.291585360842689596108595640770, −7.68104069819820777879130742532, −6.56082032331281953212474100054, −6.11024177190494655752335378466, −5.58518267755050970873880849519, −4.44854191917653877696469051060, −3.44954761732885093838302747858, −2.13439184238392726505370426224, −1.51036552432221618052807062560, 0,
1.51036552432221618052807062560, 2.13439184238392726505370426224, 3.44954761732885093838302747858, 4.44854191917653877696469051060, 5.58518267755050970873880849519, 6.11024177190494655752335378466, 6.56082032331281953212474100054, 7.68104069819820777879130742532, 8.291585360842689596108595640770
