| L(s) = 1 | + 2.97·2-s − 3·3-s + 0.824·4-s − 18.0·5-s − 8.91·6-s + 11.5·7-s − 21.3·8-s + 9·9-s − 53.7·10-s − 11·11-s − 2.47·12-s − 80.7·13-s + 34.4·14-s + 54.2·15-s − 69.9·16-s − 61.1·17-s + 26.7·18-s + 21.2·19-s − 14.8·20-s − 34.7·21-s − 32.6·22-s − 143.·23-s + 63.9·24-s + 201.·25-s − 239.·26-s − 27·27-s + 9.55·28-s + ⋯ |
| L(s) = 1 | + 1.05·2-s − 0.577·3-s + 0.103·4-s − 1.61·5-s − 0.606·6-s + 0.626·7-s − 0.942·8-s + 0.333·9-s − 1.69·10-s − 0.301·11-s − 0.0594·12-s − 1.72·13-s + 0.657·14-s + 0.933·15-s − 1.09·16-s − 0.872·17-s + 0.350·18-s + 0.256·19-s − 0.166·20-s − 0.361·21-s − 0.316·22-s − 1.30·23-s + 0.543·24-s + 1.61·25-s − 1.80·26-s − 0.192·27-s + 0.0644·28-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
| \(L(2)\) |
\(\approx\) |
\(0.007745085814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.007745085814\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.97T + 8T^{2} \) |
| 5 | \( 1 + 18.0T + 125T^{2} \) |
| 7 | \( 1 - 11.5T + 343T^{2} \) |
| 13 | \( 1 + 80.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.73T + 2.97e4T^{2} \) |
| 37 | \( 1 + 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 365.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 366.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 207.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 571.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 799.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 916.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 176.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 180.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 429.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 420.T + 9.12e5T^{2} \) |
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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
−8.682662316596064385463545007712, −7.68294207902659802287251910425, −7.35862035523089999734754451909, −6.26477533173325064805000207355, −5.23038146822756229854845960421, −4.71390661906754661183119643661, −4.11295433898690796451052585047, −3.27406345529455427478455991602, −2.05215135977244421121405040016, −0.03331439803444777819520691144,
0.03331439803444777819520691144, 2.05215135977244421121405040016, 3.27406345529455427478455991602, 4.11295433898690796451052585047, 4.71390661906754661183119643661, 5.23038146822756229854845960421, 6.26477533173325064805000207355, 7.35862035523089999734754451909, 7.68294207902659802287251910425, 8.682662316596064385463545007712
