L(s) = 1 | − 1.52·2-s − 3.31·3-s + 0.310·4-s + 0.643·5-s + 5.03·6-s − 3.98·7-s + 2.56·8-s + 7.97·9-s − 0.977·10-s − 5.02·11-s − 1.02·12-s − 13-s + 6.05·14-s − 2.13·15-s − 4.52·16-s − 1.32·17-s − 12.1·18-s + 5.39·19-s + 0.199·20-s + 13.2·21-s + 7.63·22-s − 1.98·23-s − 8.50·24-s − 4.58·25-s + 1.52·26-s − 16.4·27-s − 1.23·28-s + ⋯ |
L(s) = 1 | − 1.07·2-s − 1.91·3-s + 0.155·4-s + 0.287·5-s + 2.05·6-s − 1.50·7-s + 0.907·8-s + 2.65·9-s − 0.309·10-s − 1.51·11-s − 0.297·12-s − 0.277·13-s + 1.61·14-s − 0.550·15-s − 1.13·16-s − 0.321·17-s − 2.85·18-s + 1.23·19-s + 0.0446·20-s + 2.88·21-s + 1.62·22-s − 0.413·23-s − 1.73·24-s − 0.917·25-s + 0.298·26-s − 3.17·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04957283884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04957283884\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.52T + 2T^{2} \) |
| 3 | \( 1 + 3.31T + 3T^{2} \) |
| 5 | \( 1 - 0.643T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 + 5.02T + 11T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 + 1.98T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 - 0.282T + 41T^{2} \) |
| 43 | \( 1 - 1.38T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 2.72T + 53T^{2} \) |
| 59 | \( 1 + 0.751T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 9.59T + 67T^{2} \) |
| 71 | \( 1 - 0.812T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 6.39T + 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 + 3.54T + 97T^{2} \) |
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
−9.682901549879020116556739249931, −9.318280403700304609849538842301, −7.68237886715300452216266725446, −7.29581152767015211407726615505, −6.30108223184745084532683143323, −5.54347466907034433612419158694, −4.91175304291185106376551652796, −3.58588237268669021886686434889, −1.81408588502400900516078337492, −0.20319390225061309421418684888,
0.20319390225061309421418684888, 1.81408588502400900516078337492, 3.58588237268669021886686434889, 4.91175304291185106376551652796, 5.54347466907034433612419158694, 6.30108223184745084532683143323, 7.29581152767015211407726615505, 7.68237886715300452216266725446, 9.318280403700304609849538842301, 9.682901549879020116556739249931