| L(s) = 1 | + 2-s − 1.74·3-s + 4-s − 4.26·5-s − 1.74·6-s + 8-s + 0.0536·9-s − 4.26·10-s + 3.18·11-s − 1.74·12-s + 4.60·13-s + 7.45·15-s + 16-s − 3.15·17-s + 0.0536·18-s + 4.53·19-s − 4.26·20-s + 3.18·22-s − 23-s − 1.74·24-s + 13.1·25-s + 4.60·26-s + 5.14·27-s − 3.70·29-s + 7.45·30-s − 3.23·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.00·3-s + 0.5·4-s − 1.90·5-s − 0.713·6-s + 0.353·8-s + 0.0178·9-s − 1.34·10-s + 0.961·11-s − 0.504·12-s + 1.27·13-s + 1.92·15-s + 0.250·16-s − 0.766·17-s + 0.0126·18-s + 1.04·19-s − 0.953·20-s + 0.679·22-s − 0.208·23-s − 0.356·24-s + 2.63·25-s + 0.903·26-s + 0.990·27-s − 0.688·29-s + 1.36·30-s − 0.581·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 + 2.29T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.918T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 7.42T + 79T^{2} \) |
| 83 | \( 1 - 9.36T + 83T^{2} \) |
| 89 | \( 1 - 5.22T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−8.474034349709571790356457789113, −7.76111296878567982852987338411, −6.81200912212180053107449881470, −6.40398840285060938620696036805, −5.32652554185244990096764643280, −4.61488665570332262177510778198, −3.71167989352987190829935634332, −3.31064562961183925295589136623, −1.35324418924179083541989564013, 0,
1.35324418924179083541989564013, 3.31064562961183925295589136623, 3.71167989352987190829935634332, 4.61488665570332262177510778198, 5.32652554185244990096764643280, 6.40398840285060938620696036805, 6.81200912212180053107449881470, 7.76111296878567982852987338411, 8.474034349709571790356457789113
