L(s) = 1 | + (−1.93 − 1.11i)2-s + (1.5 + 2.59i)4-s + (−1.93 + 1.11i)5-s − 2.23i·8-s + 5.00·10-s + (0.499 − 0.866i)16-s − 4.47i·17-s − 4·19-s + (−5.80 − 3.35i)20-s + (7.74 − 4.47i)23-s + (2.5 − 4.33i)25-s + (−4 − 6.92i)31-s + (−5.80 + 3.35i)32-s + (−5.00 + 8.66i)34-s + (7.74 + 4.47i)38-s + ⋯ |
L(s) = 1 | + (−1.36 − 0.790i)2-s + (0.750 + 1.29i)4-s + (−0.866 + 0.499i)5-s − 0.790i·8-s + 1.58·10-s + (0.124 − 0.216i)16-s − 1.08i·17-s − 0.917·19-s + (−1.29 − 0.750i)20-s + (1.61 − 0.932i)23-s + (0.5 − 0.866i)25-s + (−0.718 − 1.24i)31-s + (−1.02 + 0.592i)32-s + (−0.857 + 1.48i)34-s + (1.25 + 0.725i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.163193 - 0.349969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.163193 - 0.349969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
good | 2 | \( 1 + (1.93 + 1.11i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-7.74 + 4.47i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.74 + 4.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-15.4 - 8.94i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (48.5 + 84.0i)T^{2} \) |
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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
−10.96770936579281599699642072273, −10.08466650379890213667380514949, −9.130515977637142807260696169414, −8.369814538226362840518952268733, −7.48613015698205031967928444639, −6.66135345361517135208058077696, −4.86278321638879379767305792393, −3.40819861972392036889456005048, −2.33042012379504187933894482332, −0.43020214201801468131864267480,
1.34507427513411275027406755404, 3.58117302598872744835150085828, 4.96533041134637229405097271559, 6.27365804962657593005476144460, 7.20951703150063010986622023724, 8.004893326395897102334949780281, 8.745572233669702602913182713159, 9.380647069139086456025101386553, 10.60582829174185013620351378802, 11.15982730518919094839231234710