| L(s) = 1 | + (0.500 + 1.09i)2-s + (−0.841 − 0.540i)3-s + (0.358 − 0.413i)4-s + (−0.271 − 1.88i)5-s + (0.171 − 1.19i)6-s + (−1.91 − 4.18i)7-s + (2.94 + 0.864i)8-s + (0.415 + 0.909i)9-s + (1.93 − 1.24i)10-s + (0.0433 + 0.301i)11-s + (−0.524 + 0.154i)12-s + (0.0988 − 0.0290i)13-s + (3.63 − 4.19i)14-s + (−0.791 + 1.73i)15-s + (0.371 + 2.58i)16-s + (1.23 + 1.41i)17-s + ⋯ |
| L(s) = 1 | + (0.354 + 0.775i)2-s + (−0.485 − 0.312i)3-s + (0.179 − 0.206i)4-s + (−0.121 − 0.843i)5-s + (0.0700 − 0.487i)6-s + (−0.723 − 1.58i)7-s + (1.04 + 0.305i)8-s + (0.138 + 0.303i)9-s + (0.611 − 0.392i)10-s + (0.0130 + 0.0909i)11-s + (−0.151 + 0.0444i)12-s + (0.0274 − 0.00805i)13-s + (0.971 − 1.12i)14-s + (−0.204 + 0.447i)15-s + (0.0927 + 0.645i)16-s + (0.298 + 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21013 - 0.362713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21013 - 0.362713i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (-4.94 - 6.52i)T \) |
| good | 2 | \( 1 + (-0.500 - 1.09i)T + (-1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (0.271 + 1.88i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (1.91 + 4.18i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.0433 - 0.301i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.0988 + 0.0290i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.23 - 1.41i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (1.23 - 2.71i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (-6.59 - 4.23i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + (8.64 + 2.53i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 - 9.82T + 37T^{2} \) |
| 41 | \( 1 + (-4.84 - 5.59i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (5.95 + 6.87i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (0.0939 + 0.0603i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (3.99 - 4.60i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (1.75 + 0.514i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.300 - 2.08i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (-8.02 + 9.26i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.232 + 1.61i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (3.06 - 0.899i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.834 - 5.80i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (12.3 - 7.94i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−12.84091185976136872487630843069, −11.28485464681459913544801974480, −10.53332446566338695182398824485, −9.463776848926398566628568564862, −7.83853030770670328511362009730, −7.15424923959800672645917126626, −6.17031446716237160455420018888, −5.06747710897121029715985405448, −3.93253391056084696139521222478, −1.17716669137376862163719742116,
2.52854327231894125013652539881, 3.34660275442955926668538241553, 4.97305455605980575707191361237, 6.26932925156890209426373590491, 7.21558968612367432534046284982, 8.825119087713881306427721098234, 9.815930116168746781493135567902, 11.07748670228765988059490075876, 11.33612372208096600311733081965, 12.62443335579736237755586613286
