L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.880 − 1.52i)5-s − 0.999·8-s − 1.76·10-s + (3.06 − 5.30i)11-s + (0.380 + 0.658i)13-s + (−0.5 + 0.866i)16-s + 6.84·17-s + 1.94·19-s + (−0.880 + 1.52i)20-s + (−3.06 − 5.30i)22-s + (−0.210 − 0.364i)23-s + (0.949 − 1.64i)25-s + 0.760·26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.393 − 0.681i)5-s − 0.353·8-s − 0.556·10-s + (0.923 − 1.59i)11-s + (0.105 + 0.182i)13-s + (−0.125 + 0.216i)16-s + 1.65·17-s + 0.445·19-s + (−0.196 + 0.340i)20-s + (−0.652 − 1.13i)22-s + (−0.0438 − 0.0760i)23-s + (0.189 − 0.328i)25-s + 0.149·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.073699851\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.073699851\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.880 + 1.52i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.06 + 5.30i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.380 - 0.658i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + (0.210 + 0.364i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.732 - 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.306T + 73T^{2} \) |
| 79 | \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 + (-1.81 + 3.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.744958692740211742332238320213, −8.018369382882049737264463017344, −6.99834628928488417106165383768, −5.98757561741313068117319368217, −5.42810906699085949056624312318, −4.49742349757444043070551312955, −3.56490555905005668295026486914, −3.08918064842860420899952163914, −1.47249698233592522459440435490, −0.69207241045606448748774346327,
1.38293477073359602657768335207, 2.79366350042823939024729892943, 3.64716044219094534818172631278, 4.41748851118740803216726022807, 5.28397149684227859553418196282, 6.21101163400009923235343344151, 6.88672036168087404712267337381, 7.62205681670613326806296059426, 7.973286000240538383400219274213, 9.262416086435990741992060410615