| L(s) = 1 | − 2-s − 0.803·3-s + 4-s + 5-s + 0.803·6-s + 7-s − 8-s − 2.35·9-s − 10-s − 0.803·12-s + 2.64·13-s − 14-s − 0.803·15-s + 16-s − 1.60·17-s + 2.35·18-s − 1.90·19-s + 20-s − 0.803·21-s − 4.22·23-s + 0.803·24-s + 25-s − 2.64·26-s + 4.30·27-s + 28-s − 8.97·29-s + 0.803·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.463·3-s + 0.5·4-s + 0.447·5-s + 0.327·6-s + 0.377·7-s − 0.353·8-s − 0.784·9-s − 0.316·10-s − 0.231·12-s + 0.733·13-s − 0.267·14-s − 0.207·15-s + 0.250·16-s − 0.389·17-s + 0.554·18-s − 0.438·19-s + 0.223·20-s − 0.175·21-s − 0.881·23-s + 0.163·24-s + 0.200·25-s − 0.518·26-s + 0.827·27-s + 0.188·28-s − 1.66·29-s + 0.146·30-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
| \(L(1)\) |
\(\approx\) |
\(1.007297333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007297333\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.803T + 3T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 + 4.22T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 9.69T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 4.16T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 4.58T + 71T^{2} \) |
| 73 | \( 1 + 3.69T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 3.41T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.85436319763499270042186323355, −7.17890499014761046249885097539, −6.22007155745277315136158798337, −5.96700159643613654514597828498, −5.22277828190978638381362459153, −4.27955351905739271939631447235, −3.39496131615124454827552461909, −2.38188295533834072831087588333, −1.70084507219938601940677700067, −0.55481135790589426225752656456,
0.55481135790589426225752656456, 1.70084507219938601940677700067, 2.38188295533834072831087588333, 3.39496131615124454827552461909, 4.27955351905739271939631447235, 5.22277828190978638381362459153, 5.96700159643613654514597828498, 6.22007155745277315136158798337, 7.17890499014761046249885097539, 7.85436319763499270042186323355
