L(s) = 1 | + 1.80·2-s + (−1.28 + 1.15i)3-s + 1.27·4-s + i·5-s + (−2.32 + 2.09i)6-s + i·7-s − 1.31·8-s + (0.310 − 2.98i)9-s + 1.80i·10-s + (1.44 + 2.98i)11-s + (−1.63 + 1.47i)12-s − 1.34i·13-s + 1.80i·14-s + (−1.15 − 1.28i)15-s − 4.92·16-s − 3.37·17-s + ⋯ |
L(s) = 1 | + 1.27·2-s + (−0.742 + 0.669i)3-s + 0.635·4-s + 0.447i·5-s + (−0.949 + 0.856i)6-s + 0.377i·7-s − 0.466·8-s + (0.103 − 0.994i)9-s + 0.571i·10-s + (0.434 + 0.900i)11-s + (−0.471 + 0.425i)12-s − 0.374i·13-s + 0.483i·14-s + (−0.299 − 0.332i)15-s − 1.23·16-s − 0.819·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.374023359\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.374023359\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.28 - 1.15i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-1.44 - 2.98i)T \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 13 | \( 1 + 1.34iT - 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 6.55iT - 19T^{2} \) |
| 23 | \( 1 - 0.712iT - 23T^{2} \) |
| 29 | \( 1 + 0.481T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 8.37iT - 43T^{2} \) |
| 47 | \( 1 - 8.28iT - 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 1.07iT - 59T^{2} \) |
| 61 | \( 1 - 9.12iT - 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 3.34iT - 71T^{2} \) |
| 73 | \( 1 - 8.92iT - 73T^{2} \) |
| 79 | \( 1 - 17.6iT - 79T^{2} \) |
| 83 | \( 1 + 0.931T + 83T^{2} \) |
| 89 | \( 1 - 5.94iT - 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−10.28550207698934318147987610236, −9.503154648368264231889615827121, −8.703698056145094552456194353203, −7.25792380964826204980863508587, −6.48791566333818493322753591986, −5.65790546494824509471370714589, −5.10379467082046086530998406428, −4.00790113621140968494147860796, −3.55640060490460944594543345901, −2.10121340265966156612166539895,
0.40624692755254658250750613534, 2.02215861544991435267524983509, 3.37004055604848739355732670701, 4.44274826515659498934582872420, 5.07513928364982328806610661881, 5.95875268771688824005857726390, 6.65043923954053594908036435301, 7.38991561115147940470017207499, 8.699026970256819413192208172516, 9.237362456044071675043722763028