L(s) = 1 | + (1.34 − 0.443i)2-s + (1.60 − 1.19i)4-s + (−1.27 − 1.27i)5-s − 0.158i·7-s + (1.62 − 2.31i)8-s + (−2.27 − 1.14i)10-s + (3.79 + 3.79i)11-s + (−4.21 + 4.21i)13-s + (−0.0705 − 0.213i)14-s + (1.15 − 3.82i)16-s − 3.05·17-s + (−2.15 + 2.15i)19-s + (−3.55 − 0.526i)20-s + (6.78 + 3.41i)22-s − 2.82i·23-s + ⋯ |
L(s) = 1 | + (0.949 − 0.313i)2-s + (0.803 − 0.595i)4-s + (−0.568 − 0.568i)5-s − 0.0600i·7-s + (0.575 − 0.817i)8-s + (−0.718 − 0.361i)10-s + (1.14 + 1.14i)11-s + (−1.16 + 1.16i)13-s + (−0.0188 − 0.0570i)14-s + (0.289 − 0.957i)16-s − 0.740·17-s + (−0.495 + 0.495i)19-s + (−0.795 − 0.117i)20-s + (1.44 + 0.727i)22-s − 0.589i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61475 - 0.577244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61475 - 0.577244i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.34 + 0.443i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.27 + 1.27i)T + 5iT^{2} \) |
| 7 | \( 1 + 0.158iT - 7T^{2} \) |
| 11 | \( 1 + (-3.79 - 3.79i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.21 - 4.21i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.05T + 17T^{2} \) |
| 19 | \( 1 + (2.15 - 2.15i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (2.09 - 2.09i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.15T + 31T^{2} \) |
| 37 | \( 1 + (5.98 + 5.98i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.60iT - 41T^{2} \) |
| 43 | \( 1 + (-5.75 - 5.75i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (3.55 + 3.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (4 + 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.66 + 3.66i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.767 + 0.767i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.317iT - 71T^{2} \) |
| 73 | \( 1 + 1.33iT - 73T^{2} \) |
| 79 | \( 1 + 9.69T + 79T^{2} \) |
| 83 | \( 1 + (0.115 - 0.115i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 0.571T + 97T^{2} \) |
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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
−12.63555163242934429020980341154, −12.23081601223146809349523871637, −11.36258122250695787771610312727, −10.03395915105638759775904934388, −9.001198816080772825885274298830, −7.33633704324410486709397934067, −6.46029789153683316666679001109, −4.68214791067128009684602325879, −4.14513801824832481448778993687, −2.04231455419895829426360444600,
2.85538845104502399448770315126, 4.02538472525663143726951785769, 5.49227390299227299477929295700, 6.68169759848924165227617171727, 7.62183689279022382947963565976, 8.832556977780740870368954543238, 10.53878069512776324668785992253, 11.44082002454840278184085641700, 12.19167933819502566689317636240, 13.35583193529083687310490886959