L(s) = 1 | + (−1 − 1.73i)2-s + (2.5 + 4.33i)3-s + (−1.99 + 3.46i)4-s + (6 + 10.3i)5-s + (5 − 8.66i)6-s + 8·7-s + 7.99·8-s + (0.999 − 1.73i)9-s + (12 − 20.7i)10-s + 9·11-s − 20·12-s + (−13 + 22.5i)13-s + (−8 − 13.8i)14-s + (−30.0 + 51.9i)15-s + (−8 − 13.8i)16-s + (−57 − 98.7i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.481 + 0.833i)3-s + (−0.249 + 0.433i)4-s + (0.536 + 0.929i)5-s + (0.340 − 0.589i)6-s + 0.431·7-s + 0.353·8-s + (0.0370 − 0.0641i)9-s + (0.379 − 0.657i)10-s + 0.246·11-s − 0.481·12-s + (−0.277 + 0.480i)13-s + (−0.152 − 0.264i)14-s + (−0.516 + 0.894i)15-s + (−0.125 − 0.216i)16-s + (−0.813 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.28761 + 0.278289i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28761 + 0.278289i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 19 | \( 1 + (66.5 - 49.3i)T \) |
good | 3 | \( 1 + (-2.5 - 4.33i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-6 - 10.3i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 - 9T + 1.33e3T^{2} \) |
| 13 | \( 1 + (13 - 22.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (57 + 98.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 23 | \( 1 + (-39 + 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-102 + 176. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 98T + 2.97e4T^{2} \) |
| 37 | \( 1 + 334T + 5.06e4T^{2} \) |
| 41 | \( 1 + (88.5 + 153. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-158 - 273. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-246 + 426. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (339 - 587. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-289.5 - 501. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-176 + 304. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (377.5 - 653. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-72.5 - 125. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-158 - 273. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 567T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-57 + 98.7i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-471.5 - 816. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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
−15.79344673072152904997313759790, −14.60973975809949612176928169533, −13.79477012140174774009642549862, −12.00217775923006031960173849819, −10.71420165249823358998415755773, −9.814875581255367554197956981493, −8.706774222914865863429419266714, −6.79763651794006797375943152556, −4.38639626357381357373537488245, −2.61049759785726038636040849688,
1.62139973029177866557976345889, 4.97008215825124048539788859130, 6.67426778657871342886957032830, 8.145497968852400726971723120636, 8.954429043077708115769500661252, 10.62611418324146687729592670656, 12.61974175118189989909971753823, 13.34512977626873638597211088267, 14.52408829033646498089762438057, 15.76774923889544842069056797012