| L(s) = 1 | + 5.64·2-s + 3·3-s + 23.8·4-s + 17.7·5-s + 16.9·6-s + 20.1·7-s + 89.3·8-s + 9·9-s + 100.·10-s − 11·11-s + 71.5·12-s − 46.1·13-s + 113.·14-s + 53.3·15-s + 313.·16-s − 114.·17-s + 50.7·18-s − 78.5·19-s + 423.·20-s + 60.4·21-s − 62.0·22-s − 99.6·23-s + 268.·24-s + 191.·25-s − 260.·26-s + 27·27-s + 479.·28-s + ⋯ |
| L(s) = 1 | + 1.99·2-s + 0.577·3-s + 2.97·4-s + 1.59·5-s + 1.15·6-s + 1.08·7-s + 3.94·8-s + 0.333·9-s + 3.17·10-s − 0.301·11-s + 1.72·12-s − 0.984·13-s + 2.16·14-s + 0.918·15-s + 4.89·16-s − 1.63·17-s + 0.664·18-s − 0.948·19-s + 4.73·20-s + 0.627·21-s − 0.601·22-s − 0.903·23-s + 2.27·24-s + 1.52·25-s − 1.96·26-s + 0.192·27-s + 3.23·28-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
| \(L(2)\) |
\(\approx\) |
\(16.04638738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.04638738\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.64T + 8T^{2} \) |
| 5 | \( 1 - 17.7T + 125T^{2} \) |
| 7 | \( 1 - 20.1T + 343T^{2} \) |
| 13 | \( 1 + 46.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 99.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 41.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 374.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 577.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 56.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 149.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 83.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 134.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 437.T + 9.12e5T^{2} \) |
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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
−8.727063469807647418668459469138, −7.61122794546688690894630092920, −7.02859092726489608115678437674, −6.00777955268690270676149058321, −5.60078224600370933051914408055, −4.57813227331726157592406202824, −4.23750121111584622255735226552, −2.72426515416016489857454809646, −2.14243577388014114131030907990, −1.74066887796300853401758082437,
1.74066887796300853401758082437, 2.14243577388014114131030907990, 2.72426515416016489857454809646, 4.23750121111584622255735226552, 4.57813227331726157592406202824, 5.60078224600370933051914408055, 6.00777955268690270676149058321, 7.02859092726489608115678437674, 7.61122794546688690894630092920, 8.727063469807647418668459469138
