L(s) = 1 | − 2-s + 2.79·3-s + 4-s + 3.42·5-s − 2.79·6-s + 1.08·7-s − 8-s + 4.82·9-s − 3.42·10-s + 2.08·11-s + 2.79·12-s − 2.14·13-s − 1.08·14-s + 9.56·15-s + 16-s − 4.54·17-s − 4.82·18-s − 0.688·19-s + 3.42·20-s + 3.03·21-s − 2.08·22-s − 4.23·23-s − 2.79·24-s + 6.70·25-s + 2.14·26-s + 5.09·27-s + 1.08·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.53·5-s − 1.14·6-s + 0.409·7-s − 0.353·8-s + 1.60·9-s − 1.08·10-s + 0.628·11-s + 0.807·12-s − 0.594·13-s − 0.289·14-s + 2.47·15-s + 0.250·16-s − 1.10·17-s − 1.13·18-s − 0.157·19-s + 0.765·20-s + 0.661·21-s − 0.444·22-s − 0.882·23-s − 0.570·24-s + 1.34·25-s + 0.420·26-s + 0.980·27-s + 0.204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.494993055\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.494993055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 + 0.688T + 19T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 + 0.348T + 29T^{2} \) |
| 31 | \( 1 + 6.11T + 31T^{2} \) |
| 37 | \( 1 - 2.34T + 37T^{2} \) |
| 41 | \( 1 + 8.80T + 41T^{2} \) |
| 43 | \( 1 + 4.90T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 - 9.18T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 - 3.69T + 71T^{2} \) |
| 73 | \( 1 - 0.167T + 73T^{2} \) |
| 79 | \( 1 - 9.59T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 4.13T + 89T^{2} \) |
| 97 | \( 1 - 4.88T + 97T^{2} \) |
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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
−9.886130706818787362259055082830, −9.315678278861577815603375919149, −8.785948885039996276097987182911, −7.967976113308136451017001046451, −6.99328412472064445442025508278, −6.15093562663001028649515469495, −4.81565983671438900946919317978, −3.45402378736818358350478846445, −2.16971882112667566608618000647, −1.82682223383304688089144340427,
1.82682223383304688089144340427, 2.16971882112667566608618000647, 3.45402378736818358350478846445, 4.81565983671438900946919317978, 6.15093562663001028649515469495, 6.99328412472064445442025508278, 7.967976113308136451017001046451, 8.785948885039996276097987182911, 9.315678278861577815603375919149, 9.886130706818787362259055082830