L(s) = 1 | + 2.04i·2-s − 2.19·4-s − 3.35i·5-s − 2.24i·7-s − 0.405i·8-s + 6.87·10-s + 4.93i·11-s + 4.60·14-s − 3.56·16-s + 0.911·17-s − 3.80i·19-s + 7.37i·20-s − 10.1·22-s + 2.02·23-s − 6.26·25-s + ⋯ |
L(s) = 1 | + 1.44i·2-s − 1.09·4-s − 1.50i·5-s − 0.849i·7-s − 0.143i·8-s + 2.17·10-s + 1.48i·11-s + 1.23·14-s − 0.891·16-s + 0.221·17-s − 0.872i·19-s + 1.64i·20-s − 2.15·22-s + 0.422·23-s − 1.25·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.425184398\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.425184398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.04iT - 2T^{2} \) |
| 5 | \( 1 + 3.35iT - 5T^{2} \) |
| 7 | \( 1 + 2.24iT - 7T^{2} \) |
| 11 | \( 1 - 4.93iT - 11T^{2} \) |
| 17 | \( 1 - 0.911T + 17T^{2} \) |
| 19 | \( 1 + 3.80iT - 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 + 8.80iT - 37T^{2} \) |
| 41 | \( 1 + 6.93iT - 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 3.80iT - 47T^{2} \) |
| 53 | \( 1 + 0.542T + 53T^{2} \) |
| 59 | \( 1 + 4.71iT - 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 + 2.37iT - 71T^{2} \) |
| 73 | \( 1 + 7.41iT - 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 + 2.30iT - 83T^{2} \) |
| 89 | \( 1 + 10.0iT - 89T^{2} \) |
| 97 | \( 1 + 16.1iT - 97T^{2} \) |
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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
−9.186796832176717366894981255148, −8.585068610430857655632950962427, −7.49832060190092095959797448199, −7.37579691600618376620330490931, −6.27008395588013317370378702072, −5.31546685183189055151058152079, −4.65153713391220907124012058918, −4.12480374571145881596078109018, −2.12018199270485525594339372696, −0.60581511918470796796317907006,
1.31306878585220114449510101521, 2.71658151751720849364412184373, 3.01224409845638604410834341729, 3.85093449549676267761320583250, 5.26342811765687557994664339789, 6.25169915444579243676892540764, 6.86932569193475469486993256826, 8.159792087714794620653892370460, 8.821270877096831360108989148899, 9.827413637965662807462558523519