L(s) = 1 | + (−0.258 − 0.965i)2-s + (2.51 − 0.672i)3-s + (−0.866 + 0.499i)4-s + (−1.29 − 2.25i)6-s + (−0.692 − 0.692i)7-s + (0.707 + 0.707i)8-s + (3.25 − 1.87i)9-s + 4.06·11-s + (−1.83 + 1.83i)12-s + (1.66 − 6.19i)13-s + (−0.489 + 0.847i)14-s + (0.500 − 0.866i)16-s + (−2.50 + 0.670i)17-s + (−2.65 − 2.65i)18-s + (0.843 + 4.27i)19-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (1.44 − 0.388i)3-s + (−0.433 + 0.249i)4-s + (−0.530 − 0.918i)6-s + (−0.261 − 0.261i)7-s + (0.249 + 0.249i)8-s + (1.08 − 0.625i)9-s + 1.22·11-s + (−0.530 + 0.530i)12-s + (0.460 − 1.71i)13-s + (−0.130 + 0.226i)14-s + (0.125 − 0.216i)16-s + (−0.606 + 0.162i)17-s + (−0.625 − 0.625i)18-s + (0.193 + 0.981i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0398 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0398 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61401 - 1.67974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61401 - 1.67974i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-0.843 - 4.27i)T \) |
good | 3 | \( 1 + (-2.51 + 0.672i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.692 + 0.692i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + (-1.66 + 6.19i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (2.50 - 0.670i)T + (14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-3.52 - 0.943i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.89 + 3.28i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.30iT - 31T^{2} \) |
| 37 | \( 1 + (-2.58 + 2.58i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.09 + 4.09i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.23 + 4.61i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.68 + 6.29i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.565 - 2.10i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.99 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.43 + 7.68i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.25 - 2.47i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.72 - 3.88i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.02 - 15.0i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.49 - 6.05i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.44 + 4.44i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.61 - 2.80i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.62 + 9.80i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−9.778733031831504551984399654919, −8.868890725455654735708304658329, −8.411154690349787289092611344994, −7.57703987477545130242699486839, −6.66962166360148775306039560185, −5.37971251547618863392684726294, −3.75739275268919080833030392959, −3.47769336574046991817949965383, −2.27973556417531075061569666217, −1.12195960009795238921813127840,
1.68559887319350803560865917010, 2.99350940194915525428490148109, 4.06224544249351877615169134737, 4.71842767958999882755473872173, 6.36176649145341486387920207073, 6.82832408233486029774412178133, 7.88632219748923667920733455010, 8.846888466121467514933065599915, 9.251139963569640848015893930430, 9.551940828423274106143796296485