L(s) = 1 | + (3.03 − 4.77i)2-s − 23.6·3-s + (−13.5 − 28.9i)4-s + (−71.7 + 112. i)6-s + 160. i·7-s + (−179. − 23.4i)8-s + 314.·9-s − 129. i·11-s + (319. + 684. i)12-s + 759.·13-s + (766. + 488. i)14-s + (−657. + 785. i)16-s + 323. i·17-s + (955. − 1.50e3i)18-s − 198. i·19-s + ⋯ |
L(s) = 1 | + (0.537 − 0.843i)2-s − 1.51·3-s + (−0.423 − 0.906i)4-s + (−0.813 + 1.27i)6-s + 1.23i·7-s + (−0.991 − 0.129i)8-s + 1.29·9-s − 0.321i·11-s + (0.641 + 1.37i)12-s + 1.24·13-s + (1.04 + 0.665i)14-s + (−0.641 + 0.766i)16-s + 0.271i·17-s + (0.694 − 1.09i)18-s − 0.126i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.057610142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.057610142\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.03 + 4.77i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 23.6T + 243T^{2} \) |
| 7 | \( 1 - 160. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 129. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 759.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 323. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 198. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.19e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.98e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.69e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.04e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.87e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.35e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.85e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.12e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.37e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.25e4iT - 8.58e9T^{2} \) |
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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
−11.41637206338824983896004801419, −10.72135896235184130891089468380, −9.586260465007933732388494846752, −8.478920549827805365274689143824, −6.40771621911847821791446893386, −5.80837380629824899490847258731, −4.99389689761302644042274935914, −3.56089671973284099074275978843, −1.86975325794326641026882816847, −0.44022497863143795405037164598,
0.944422532740760149998556910518, 3.67671678225441656285880280395, 4.68315448563793631038728434550, 5.69373039698848417842748634514, 6.66641943925685524549875271589, 7.34269566585559942525691747840, 8.724731470689100747741009098118, 10.25718524090950503818541845904, 11.05119202230623792000650351620, 11.97247456744943870407378387559