| L(s) = 1 | + 1.43·2-s + 3·3-s − 5.95·4-s + 11.9·5-s + 4.29·6-s − 33.6·7-s − 19.9·8-s + 9·9-s + 17.1·10-s − 11·11-s − 17.8·12-s − 12.5·13-s − 48.1·14-s + 35.9·15-s + 19.0·16-s − 74.2·17-s + 12.8·18-s − 68.3·19-s − 71.3·20-s − 100.·21-s − 15.7·22-s + 60.0·23-s − 59.9·24-s + 18.9·25-s − 17.9·26-s + 27·27-s + 200.·28-s + ⋯ |
| L(s) = 1 | + 0.506·2-s + 0.577·3-s − 0.743·4-s + 1.07·5-s + 0.292·6-s − 1.81·7-s − 0.882·8-s + 0.333·9-s + 0.543·10-s − 0.301·11-s − 0.429·12-s − 0.267·13-s − 0.919·14-s + 0.619·15-s + 0.297·16-s − 1.05·17-s + 0.168·18-s − 0.825·19-s − 0.798·20-s − 1.04·21-s − 0.152·22-s + 0.544·23-s − 0.509·24-s + 0.151·25-s − 0.135·26-s + 0.192·27-s + 1.35·28-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
| \(L(2)\) |
\(\approx\) |
\(1.855158810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855158810\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.43T + 8T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 + 33.6T + 343T^{2} \) |
| 13 | \( 1 + 12.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 60.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 5.95T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 503.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 357.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 932.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 553.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 790.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.037244138423997768372740495563, −8.259283392419644500476399159863, −6.90383672896042741290225505322, −6.38801540843132391894211114761, −5.65055036169887440828365321564, −4.68366666968778876852396454342, −3.78818247691695650456937953089, −2.92554405725339910638098609854, −2.22883806188609817706113756390, −0.53537683900761541280791377400,
0.53537683900761541280791377400, 2.22883806188609817706113756390, 2.92554405725339910638098609854, 3.78818247691695650456937953089, 4.68366666968778876852396454342, 5.65055036169887440828365321564, 6.38801540843132391894211114761, 6.90383672896042741290225505322, 8.259283392419644500476399159863, 9.037244138423997768372740495563
