| L(s) = 1 | − 7.96·7-s + 44.6·11-s + 68.4·13-s + 90.1·17-s + 75.3·19-s − 11.0·23-s − 145.·29-s + 202.·31-s − 124.·37-s + 150.·41-s − 134.·43-s − 510.·47-s − 279.·49-s − 16.2·53-s + 399.·59-s − 486.·61-s + 353.·67-s − 76.9·71-s − 419.·73-s − 355.·77-s + 110.·79-s + 45.6·83-s + 724.·89-s − 545.·91-s + 539.·97-s + 1.48e3·101-s + 1.20e3·103-s + ⋯ |
| L(s) = 1 | − 0.430·7-s + 1.22·11-s + 1.45·13-s + 1.28·17-s + 0.910·19-s − 0.100·23-s − 0.934·29-s + 1.17·31-s − 0.555·37-s + 0.575·41-s − 0.476·43-s − 1.58·47-s − 0.814·49-s − 0.0422·53-s + 0.880·59-s − 1.02·61-s + 0.644·67-s − 0.128·71-s − 0.673·73-s − 0.526·77-s + 0.157·79-s + 0.0603·83-s + 0.863·89-s − 0.627·91-s + 0.564·97-s + 1.46·101-s + 1.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.727824041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.727824041\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7.96T + 343T^{2} \) |
| 11 | \( 1 - 44.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 90.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 134.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 510.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 16.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 353.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 76.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 419.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 110.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 45.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 724.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 539.T + 9.12e5T^{2} \) |
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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
−8.948770548473407817900034735782, −8.183409403509704536501974458430, −7.32883348749769915038086327792, −6.37667009455397306302831826709, −5.90798655457927519955178084242, −4.82126030002498579256222116428, −3.63044896460336382155919991571, −3.28725476975544639642116869050, −1.64814644715870697835120250597, −0.845312683783867828613923088854,
0.845312683783867828613923088854, 1.64814644715870697835120250597, 3.28725476975544639642116869050, 3.63044896460336382155919991571, 4.82126030002498579256222116428, 5.90798655457927519955178084242, 6.37667009455397306302831826709, 7.32883348749769915038086327792, 8.183409403509704536501974458430, 8.948770548473407817900034735782
