| L(s) = 1 | − 63.6i·5-s − 524·7-s − 865. i·11-s − 344·13-s − 7.14e3i·17-s − 2.32e3·19-s − 5.75e3i·23-s + 1.15e4·25-s − 2.31e4i·29-s + 1.05e4·31-s + 3.33e4i·35-s + 2.40e4·37-s − 1.08e5i·41-s − 9.09e4·43-s − 1.28e5i·47-s + ⋯ |
| L(s) = 1 | − 0.509i·5-s − 1.52·7-s − 0.650i·11-s − 0.156·13-s − 1.45i·17-s − 0.338·19-s − 0.472i·23-s + 0.740·25-s − 0.949i·29-s + 0.354·31-s + 0.777i·35-s + 0.475·37-s − 1.57i·41-s − 1.14·43-s − 1.24i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.2943469367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2943469367\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 63.6iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 524T + 1.17e5T^{2} \) |
| 11 | \( 1 + 865. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 344T + 4.82e6T^{2} \) |
| 17 | \( 1 + 7.14e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 2.32e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 5.75e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.31e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.05e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 1.08e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.09e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.28e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.96e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.98e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.51e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.16e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.39e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 3.08e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.40e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.32e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.23e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.71e4T + 8.32e11T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.227282849030912695823453915701, −8.505039694131558134458698000156, −7.27005531441380484231783399801, −6.48319384859896210725075122432, −5.58181394557847400315367555228, −4.49038605577193523331668425286, −3.33061086079824348267164159869, −2.49888002446713561882238085546, −0.77607859000474277988439676141, −0.07814800974230791379961315150,
1.50813583654701782601079964628, 2.83395564494553859431208854810, 3.57141848465705096954004214275, 4.74392314866871536944431509657, 6.15176433891682660740565559887, 6.58254158948280514691330919801, 7.55753728883391248111323558358, 8.665236268987532315821820592657, 9.640331887996294713951347125292, 10.21503475444558190708530263731
