L(s) = 1 | − 2-s − 2.01·3-s + 4-s + 3.52·5-s + 2.01·6-s − 0.0194·7-s − 8-s + 1.07·9-s − 3.52·10-s + 2.98·11-s − 2.01·12-s + 6.73·13-s + 0.0194·14-s − 7.11·15-s + 16-s − 5.56·17-s − 1.07·18-s − 2.82·19-s + 3.52·20-s + 0.0393·21-s − 2.98·22-s − 5.57·23-s + 2.01·24-s + 7.40·25-s − 6.73·26-s + 3.88·27-s − 0.0194·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.16·3-s + 0.5·4-s + 1.57·5-s + 0.824·6-s − 0.00736·7-s − 0.353·8-s + 0.359·9-s − 1.11·10-s + 0.899·11-s − 0.582·12-s + 1.86·13-s + 0.00520·14-s − 1.83·15-s + 0.250·16-s − 1.34·17-s − 0.254·18-s − 0.647·19-s + 0.787·20-s + 0.00858·21-s − 0.636·22-s − 1.16·23-s + 0.412·24-s + 1.48·25-s − 1.32·26-s + 0.746·27-s − 0.00368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 2.01T + 3T^{2} \) |
| 5 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 + 0.0194T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 6.73T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.57T + 23T^{2} \) |
| 29 | \( 1 + 0.961T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 - 0.282T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 1.77T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.456010547926440761702367489039, −7.00397885623288670083747108561, −6.26146654256634385342004606455, −6.22361553427903320717868721418, −5.44800962982257648344435008971, −4.41718018078608577306914915379, −3.33353613362134572972747849553, −1.87153151658795068423817478923, −1.51027735291737635723691479297, 0,
1.51027735291737635723691479297, 1.87153151658795068423817478923, 3.33353613362134572972747849553, 4.41718018078608577306914915379, 5.44800962982257648344435008971, 6.22361553427903320717868721418, 6.26146654256634385342004606455, 7.00397885623288670083747108561, 8.456010547926440761702367489039