L(s) = 1 | − 0.960i·2-s − 5.19i·3-s + 15.0·4-s − 34.5i·5-s − 4.99·6-s + 54.1i·7-s − 29.8i·8-s − 27·9-s − 33.2·10-s − 144. i·11-s − 78.3i·12-s − 136. i·13-s + 52.0·14-s − 179.·15-s + 212.·16-s − 424.·17-s + ⋯ |
L(s) = 1 | − 0.240i·2-s − 0.577i·3-s + 0.942·4-s − 1.38i·5-s − 0.138·6-s + 1.10i·7-s − 0.466i·8-s − 0.333·9-s − 0.332·10-s − 1.19i·11-s − 0.544i·12-s − 0.807i·13-s + 0.265·14-s − 0.798·15-s + 0.830·16-s − 1.46·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.012549501\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.012549501\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19iT \) |
| 67 | \( 1 + (2.62e3 + 3.64e3i)T \) |
good | 2 | \( 1 + 0.960iT - 16T^{2} \) |
| 5 | \( 1 + 34.5iT - 625T^{2} \) |
| 7 | \( 1 - 54.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 144. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 136. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 424.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 661.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 351.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 561.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 942. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 840.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.06e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.10e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 963.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 2.78e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 234.T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.34e3iT - 1.38e7T^{2} \) |
| 71 | \( 1 - 1.79e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.59e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.55e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.16e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.32e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4iT - 8.85e7T^{2} \) |
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
−11.75621783328898516166604499642, −10.63228419588157508213345553064, −9.096560943198415238959781971482, −8.511345135130955324806101354763, −7.35903927670723157124533586179, −5.93675348690981775175259300684, −5.31101203348044035938800370109, −3.27432571670403063576212328683, −1.95860117147999020071023548943, −0.67000423278320704838813591381,
2.00645563544195939209758700170, 3.30775135801531107370535335842, 4.55631895314306527057170448153, 6.27722472385595792643586255284, 7.07245341685735423890625837938, 7.66598254793295808261034865555, 9.587052782191579145277669503249, 10.26347990738406501666469585012, 11.23116666600902646939077984227, 11.64154174993570891075547446485