L(s) = 1 | − 5.38i·2-s − 20.9·4-s + (5.57 − 9.66i)5-s + (−8.46 − 16.4i)7-s + 69.7i·8-s + (−52.0 − 30.0i)10-s + (−18.1 + 10.4i)11-s + (−46.0 + 26.5i)13-s + (−88.6 + 45.5i)14-s + 207.·16-s + (44.2 − 76.6i)17-s + (−40.4 + 23.3i)19-s + (−116. + 202. i)20-s + (56.2 + 97.4i)22-s + (68.0 + 39.3i)23-s + ⋯ |
L(s) = 1 | − 1.90i·2-s − 2.61·4-s + (0.499 − 0.864i)5-s + (−0.457 − 0.889i)7-s + 3.08i·8-s + (−1.64 − 0.949i)10-s + (−0.496 + 0.286i)11-s + (−0.981 + 0.566i)13-s + (−1.69 + 0.869i)14-s + 3.24·16-s + (0.631 − 1.09i)17-s + (−0.488 + 0.281i)19-s + (−1.30 + 2.26i)20-s + (0.545 + 0.944i)22-s + (0.617 + 0.356i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.526413 + 0.336764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.526413 + 0.336764i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (8.46 + 16.4i)T \) |
good | 2 | \( 1 + 5.38iT - 8T^{2} \) |
| 5 | \( 1 + (-5.57 + 9.66i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (18.1 - 10.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (46.0 - 26.5i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-44.2 + 76.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.4 - 23.3i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-68.0 - 39.3i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-17.2 - 9.97i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 28.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (122. + 211. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (16.9 + 29.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-87.5 + 151. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (656. + 378. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 83.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 92.1iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 897.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 706. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (529. + 305. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + 266.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-311. + 539. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (355. + 615. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (498. + 287. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.33147086489494640040642428438, −10.25467589797928947337454722375, −9.657710402432446796192587900921, −8.887014098001496937078693628797, −7.39558442019032762157396374343, −5.25541860458768007000837407768, −4.43758844896198632683869674035, −3.05904359855241995509141597427, −1.64719529226126339664482031152, −0.27871972516137515001819013207,
3.00497793710117694183808895966, 4.86829538936372393819203936109, 5.91817922743630025656918138148, 6.53511090911216206266370001633, 7.68361878587691636332951407717, 8.586334641589451482670176633282, 9.663913239812329405866250630727, 10.50401391168042064607675156490, 12.44239620660459778735987896537, 13.16631859728325649258673224102