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
−9.262416086435990741992060410615, −7.973286000240538383400219274213, −7.62205681670613326806296059426, −6.88672036168087404712267337381, −6.21101163400009923235343344151, −5.28397149684227859553418196282, −4.41748851118740803216726022807, −3.64716044219094534818172631278, −2.79366350042823939024729892943, −1.38293477073359602657768335207,
0.69207241045606448748774346327, 1.47249698233592522459440435490, 3.08918064842860420899952163914, 3.56490555905005668295026486914, 4.49742349757444043070551312955, 5.42810906699085949056624312318, 5.98757561741313068117319368217, 6.99834628928488417106165383768, 8.018369382882049737264463017344, 8.744958692740211742332238320213