| L(s) = 1 | + (−19.3 − 24.2i)2-s + (−19.7 − 2.98i)3-s + (−99.3 + 435. i)4-s + (−962. − 656. i)5-s + (309. + 536. i)6-s + (642. − 1.11e3i)7-s + (−1.82e3 + 880. i)8-s + (−1.84e4 − 5.68e3i)9-s + (2.69e3 + 3.59e4i)10-s + (−1.05e4 − 4.60e4i)11-s + (3.26e3 − 8.31e3i)12-s + (−1.09e4 + 1.45e5i)13-s + (−3.93e4 + 5.92e3i)14-s + (1.70e4 + 1.58e4i)15-s + (2.62e5 + 1.26e5i)16-s + (−2.21e5 + 1.50e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.853 − 1.06i)2-s + (−0.141 − 0.0212i)3-s + (−0.194 + 0.850i)4-s + (−0.689 − 0.469i)5-s + (0.0975 + 0.168i)6-s + (0.101 − 0.175i)7-s + (−0.157 + 0.0759i)8-s + (−0.936 − 0.288i)9-s + (0.0852 + 1.13i)10-s + (−0.216 − 0.948i)11-s + (0.0454 − 0.115i)12-s + (−0.106 + 1.41i)13-s + (−0.273 + 0.0412i)14-s + (0.0871 + 0.0808i)15-s + (1.00 + 0.482i)16-s + (−0.641 + 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.381502 - 0.0424703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.381502 - 0.0424703i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.41e6 + 2.21e7i)T \) |
| good | 2 | \( 1 + (19.3 + 24.2i)T + (-113. + 499. i)T^{2} \) |
| 3 | \( 1 + (19.7 + 2.98i)T + (1.88e4 + 5.80e3i)T^{2} \) |
| 5 | \( 1 + (962. + 656. i)T + (7.13e5 + 1.81e6i)T^{2} \) |
| 7 | \( 1 + (-642. + 1.11e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.05e4 + 4.60e4i)T + (-2.12e9 + 1.02e9i)T^{2} \) |
| 13 | \( 1 + (1.09e4 - 1.45e5i)T + (-1.04e10 - 1.58e9i)T^{2} \) |
| 17 | \( 1 + (2.21e5 - 1.50e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (4.77e5 - 1.47e5i)T + (2.66e11 - 1.81e11i)T^{2} \) |
| 23 | \( 1 + (-6.49e5 + 6.02e5i)T + (1.34e11 - 1.79e12i)T^{2} \) |
| 29 | \( 1 + (-4.51e6 + 6.81e5i)T + (1.38e13 - 4.27e12i)T^{2} \) |
| 31 | \( 1 + (-2.53e6 + 6.45e6i)T + (-1.93e13 - 1.79e13i)T^{2} \) |
| 37 | \( 1 + (-9.90e6 - 1.71e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 + (1.79e7 + 2.24e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 47 | \( 1 + (7.82e5 - 3.42e6i)T + (-1.00e15 - 4.85e14i)T^{2} \) |
| 53 | \( 1 + (-5.69e6 - 7.59e7i)T + (-3.26e15 + 4.91e14i)T^{2} \) |
| 59 | \( 1 + (-7.59e7 - 3.65e7i)T + (5.40e15 + 6.77e15i)T^{2} \) |
| 61 | \( 1 + (-4.36e7 - 1.11e8i)T + (-8.57e15 + 7.95e15i)T^{2} \) |
| 67 | \( 1 + (7.51e7 - 2.31e7i)T + (2.24e16 - 1.53e16i)T^{2} \) |
| 71 | \( 1 + (1.42e8 + 1.31e8i)T + (3.42e15 + 4.57e16i)T^{2} \) |
| 73 | \( 1 + (-1.63e7 + 2.18e8i)T + (-5.82e16 - 8.77e15i)T^{2} \) |
| 79 | \( 1 + (3.12e8 - 5.41e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-3.75e8 - 5.66e7i)T + (1.78e17 + 5.51e16i)T^{2} \) |
| 89 | \( 1 + (1.32e7 + 1.99e6i)T + (3.34e17 + 1.03e17i)T^{2} \) |
| 97 | \( 1 + (-2.57e7 - 1.12e8i)T + (-6.84e17 + 3.29e17i)T^{2} \) |
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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
−13.74174624604170961998037848669, −12.12205503324247404511315867444, −11.50997429335408167913139706384, −10.50196830288745369185463642754, −8.912416241962367367172523045015, −8.343783084597390264049505594135, −6.22404504976354215091085082887, −4.17400809255429217324321760918, −2.52041971076030490938221174726, −0.805328615532294070390004313224,
0.26449159265570953251420551011, 2.92612057560425497122016577304, 5.15997845611540628114789973855, 6.66491136261425175431264018341, 7.76402099157975372706757683526, 8.665035720432342970544895373408, 10.17567633221806140099371030537, 11.44514798215432115965230286280, 12.82624657071406567102153108012, 14.66594363255951903095254096943
