L(s) = 1 | + 2.10·2-s − 3·3-s − 3.56·4-s + 6.12·5-s − 6.31·6-s − 19.2·7-s − 24.3·8-s + 9·9-s + 12.8·10-s + 11·11-s + 10.7·12-s + 47.8·13-s − 40.6·14-s − 18.3·15-s − 22.7·16-s + 35.6·17-s + 18.9·18-s − 138.·19-s − 21.8·20-s + 57.8·21-s + 23.1·22-s + 16.0·23-s + 73.0·24-s − 87.4·25-s + 100.·26-s − 27·27-s + 68.8·28-s + ⋯ |
L(s) = 1 | + 0.744·2-s − 0.577·3-s − 0.446·4-s + 0.547·5-s − 0.429·6-s − 1.04·7-s − 1.07·8-s + 0.333·9-s + 0.407·10-s + 0.301·11-s + 0.257·12-s + 1.02·13-s − 0.775·14-s − 0.316·15-s − 0.354·16-s + 0.508·17-s + 0.248·18-s − 1.67·19-s − 0.244·20-s + 0.601·21-s + 0.224·22-s + 0.145·23-s + 0.621·24-s − 0.699·25-s + 0.759·26-s − 0.192·27-s + 0.464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.419583512\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419583512\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
good | 2 | \( 1 - 2.10T + 8T^{2} \) |
| 5 | \( 1 - 6.12T + 125T^{2} \) |
| 7 | \( 1 + 19.2T + 343T^{2} \) |
| 13 | \( 1 - 47.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 138.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 16.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 218.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 407.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 386.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 197.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 941.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 105.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 122.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 721.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 152.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 707.T + 9.12e5T^{2} \) |
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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
−8.900951754867629157732557415497, −8.131935045432382275141756193313, −6.70642729555330081870048122251, −6.26235523309264650212328569595, −5.71024920499428654704679347615, −4.77528640024268021916396985432, −3.86820729059032672869848258485, −3.24669329043654191135561260723, −1.88273265575047182330171583639, −0.49543562071670982845137841051,
0.49543562071670982845137841051, 1.88273265575047182330171583639, 3.24669329043654191135561260723, 3.86820729059032672869848258485, 4.77528640024268021916396985432, 5.71024920499428654704679347615, 6.26235523309264650212328569595, 6.70642729555330081870048122251, 8.131935045432382275141756193313, 8.900951754867629157732557415497