| L(s) = 1 | − 2.70i·5-s − 4.43i·7-s − 5.82·13-s − 4.56i·17-s − 2.91i·19-s − 5.46·23-s − 2.31·25-s − 0.847i·29-s + 4.04i·31-s − 11.9·35-s + 37-s + 4.24i·41-s + 1.77i·43-s − 7.87·47-s − 12.6·49-s + ⋯ |
| L(s) = 1 | − 1.20i·5-s − 1.67i·7-s − 1.61·13-s − 1.10i·17-s − 0.667i·19-s − 1.14·23-s − 0.462·25-s − 0.157i·29-s + 0.726i·31-s − 2.02·35-s + 0.164·37-s + 0.662i·41-s + 0.271i·43-s − 1.14·47-s − 1.80·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7348571019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7348571019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 2.70iT - 5T^{2} \) |
| 7 | \( 1 + 4.43iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5.82T + 13T^{2} \) |
| 17 | \( 1 + 4.56iT - 17T^{2} \) |
| 19 | \( 1 + 2.91iT - 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 0.847iT - 29T^{2} \) |
| 31 | \( 1 - 4.04iT - 31T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 1.77iT - 43T^{2} \) |
| 47 | \( 1 + 7.87T + 47T^{2} \) |
| 53 | \( 1 + 1.16iT - 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 + 12.5iT - 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 9.31T + 73T^{2} \) |
| 79 | \( 1 + 5.95iT - 79T^{2} \) |
| 83 | \( 1 - 7.87T + 83T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−7.71285004901168833675391082852, −7.11128676129190040703346170518, −6.46956295122400449556818271319, −5.14948628729257234549541529952, −4.83102526961103802169609047319, −4.21708276430240809701333571141, −3.23592794286657682377672948217, −2.11211867296657598430167797076, −0.950227733980928433353419035340, −0.21493603656564473539165658818,
2.01433434474459252285016885469, 2.37430087547477811302972958479, 3.24270901154451185730756537342, 4.14531770632990586285833363350, 5.24262974724251007966053584807, 5.81346226771603160313514431455, 6.44360665314445069960887121577, 7.18390843482350884741465600801, 7.994220401497586072281182131527, 8.506473096177139480293335097895
