| L(s) = 1 | + (8 − 13.8i)2-s + (−89.6 − 155. i)3-s + (−127. − 221. i)4-s + (84.1 − 145. i)5-s − 2.86e3·6-s + (−3.16e3 + 5.50e3i)7-s − 4.09e3·8-s + (−6.23e3 + 1.08e4i)9-s + (−1.34e3 − 2.33e3i)10-s + (−7.24e3 − 1.25e4i)11-s + (−2.29e4 + 3.97e4i)12-s − 1.09e5·13-s + (5.10e4 + 8.78e4i)14-s − 3.01e4·15-s + (−3.27e4 + 5.67e4i)16-s + (−1.50e5 − 2.60e5i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.639 − 1.10i)3-s + (−0.249 − 0.433i)4-s + (0.0602 − 0.104i)5-s − 0.903·6-s + (−0.497 + 0.867i)7-s − 0.353·8-s + (−0.316 + 0.548i)9-s + (−0.0425 − 0.0737i)10-s + (−0.149 − 0.258i)11-s + (−0.319 + 0.553i)12-s − 1.05·13-s + (0.355 + 0.611i)14-s − 0.153·15-s + (−0.125 + 0.216i)16-s + (−0.436 − 0.755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.186585 + 0.801089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.186585 + 0.801089i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 + 13.8i)T \) |
| 7 | \( 1 + (3.16e3 - 5.50e3i)T \) |
| good | 3 | \( 1 + (89.6 + 155. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-84.1 + 145. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (7.24e3 + 1.25e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 1.09e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.50e5 + 2.60e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-2.45e5 + 4.25e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.13e6 + 1.96e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 + 3.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (3.95e6 + 6.84e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (7.07e6 - 1.22e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.11e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.50e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-2.18e7 + 3.78e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.75e7 - 4.77e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.45e7 - 4.24e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-7.09e7 + 1.22e8i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-4.93e7 - 8.55e7i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 - 2.96e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.30e7 + 3.98e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-2.76e8 + 4.78e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 5.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.54e8 - 4.40e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 + 1.18e9T + 7.60e17T^{2} \) |
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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
−16.93088539659932734722788398150, −15.08854496726686250089261560094, −13.27667195426621326326058538276, −12.42168376264882418754394086192, −11.31750315039536789136339009071, −9.279807580912588274149717239787, −6.95698128348836074955227327666, −5.35058363748220981566030719703, −2.46515160371259556897164843742, −0.41326273800254849928972115574,
3.88438116571029492503093097543, 5.36474338217687532026267019500, 7.24320180417348290639237061150, 9.601749549373528873095706400798, 10.82359483439990403313940796765, 12.70709501385833257501569440107, 14.36539391524057623149532712325, 15.66925149633858078199739254759, 16.66108590903463750893467166510, 17.55788065520268764296420500134
