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 | \( 1 + (-471.5 + 816. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−15.76774923889544842069056797012, −14.52408829033646498089762438057, −13.34512977626873638597211088267, −12.61974175118189989909971753823, −10.62611418324146687729592670656, −8.954429043077708115769500661252, −8.145497968852400726971723120636, −6.67426778657871342886957032830, −4.97008215825124048539788859130, −1.62139973029177866557976345889,
2.61049759785726038636040849688, 4.38639626357381357373537488245, 6.79763651794006797375943152556, 8.706774222914865863429419266714, 9.814875581255367554197956981493, 10.71420165249823358998415755773, 12.00217775923006031960173849819, 13.79477012140174774009642549862, 14.60973975809949612176928169533, 15.79344673072152904997313759790