| L(s) = 1 | + (−1.16 + 0.804i)2-s + (0.564 + 0.898i)3-s + (0.705 − 1.87i)4-s + (1.84 + 1.47i)5-s + (−1.37 − 0.591i)6-s + (−0.762 − 0.174i)7-s + (0.684 + 2.74i)8-s + (0.812 − 1.68i)9-s + (−3.33 − 0.227i)10-s + (−1.41 + 4.03i)11-s + (2.08 − 0.422i)12-s + (1.14 + 2.38i)13-s + (1.02 − 0.410i)14-s + (−0.281 + 2.49i)15-s + (−3.00 − 2.64i)16-s + (−0.864 − 0.864i)17-s + ⋯ |
| L(s) = 1 | + (−0.822 + 0.568i)2-s + (0.326 + 0.518i)3-s + (0.352 − 0.935i)4-s + (0.827 + 0.659i)5-s + (−0.563 − 0.241i)6-s + (−0.288 − 0.0657i)7-s + (0.241 + 0.970i)8-s + (0.270 − 0.562i)9-s + (−1.05 − 0.0720i)10-s + (−0.425 + 1.21i)11-s + (0.600 − 0.121i)12-s + (0.318 + 0.661i)13-s + (0.274 − 0.109i)14-s + (−0.0725 + 0.644i)15-s + (−0.750 − 0.660i)16-s + (−0.209 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.701036 + 0.527760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.701036 + 0.527760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 - 0.804i)T \) |
| 29 | \( 1 + (-1.27 + 5.23i)T \) |
| good | 3 | \( 1 + (-0.564 - 0.898i)T + (-1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (-1.84 - 1.47i)T + (1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (0.762 + 0.174i)T + (6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (1.41 - 4.03i)T + (-8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-1.14 - 2.38i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (0.864 + 0.864i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.53 + 2.22i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-5.08 + 4.05i)T + (5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (0.840 + 7.45i)T + (-30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (3.11 - 1.08i)T + (28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (6.22 - 6.22i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.71 + 0.305i)T + (41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-5.74 - 2.00i)T + (36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (1.82 - 2.28i)T + (-11.7 - 51.6i)T^{2} \) |
| 59 | \( 1 + 3.52iT - 59T^{2} \) |
| 61 | \( 1 + (-6.36 + 4.00i)T + (26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 5.23i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (7.65 - 3.68i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (2.76 + 0.311i)T + (71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 3.68i)T + (61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (17.0 - 3.88i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (13.9 - 1.57i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (4.13 + 2.59i)T + (42.0 + 87.3i)T^{2} \) |
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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
−14.11926456725869989170391539763, −12.92217070329962874828525029056, −11.28853744099421167444229433059, −10.08996925454180454703811116230, −9.702018954801347792151291149133, −8.607387127109874610523720430013, −7.00158387666845721695779357201, −6.34957242842597619393086462896, −4.60303584300196884709619100259, −2.36837339885232785004602175885,
1.54871789962198585255777225868, 3.18232159344914956729683674374, 5.41401325356783867541092137785, 6.95739406794382231136988510788, 8.328712908374308859427281349130, 8.872973220129928557201296261778, 10.24878720832215296415009322687, 10.97766963285364853232018363224, 12.56292021452110653806912768133, 13.10863129141808704052837252379
