| L(s) = 1 | + 5.38i·3-s − 4·5-s − 7-s − 19.9·9-s + 14·11-s + 16.1i·13-s − 21.5i·15-s + 23·17-s + (10 + 16.1i)19-s − 5.38i·21-s − 23-s − 9·25-s − 59.2i·27-s − 48.4i·29-s − 32.3i·31-s + ⋯ |
| L(s) = 1 | + 1.79i·3-s − 0.800·5-s − 0.142·7-s − 2.22·9-s + 1.27·11-s + 1.24i·13-s − 1.43i·15-s + 1.35·17-s + (0.526 + 0.850i)19-s − 0.256i·21-s − 0.0434·23-s − 0.359·25-s − 2.19i·27-s − 1.67i·29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.518043 + 0.929917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518043 + 0.929917i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-10 - 16.1i)T \) |
| good | 3 | \( 1 - 5.38iT - 9T^{2} \) |
| 5 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 + T + 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 - 16.1iT - 169T^{2} \) |
| 17 | \( 1 - 23T + 289T^{2} \) |
| 23 | \( 1 + T + 529T^{2} \) |
| 29 | \( 1 + 48.4iT - 841T^{2} \) |
| 31 | \( 1 + 32.3iT - 961T^{2} \) |
| 37 | \( 1 - 32.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 68T + 1.84e3T^{2} \) |
| 47 | \( 1 - 26T + 2.20e3T^{2} \) |
| 53 | \( 1 + 80.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 32T + 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.9iT - 9.40e3T^{2} \) |
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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
−14.73792726753158658111641539903, −14.06909735597060663396202096686, −11.86287284820905195503533809962, −11.46964466416911543284905783279, −9.928181602485814240881104085420, −9.353801653649304380165595301930, −7.966893195597801610108853043856, −6.01662473603633316259496512174, −4.35706435694904317520744705856, −3.62646101573266471213030002533,
1.02410756581701884745227159159, 3.23531562519120511534305102665, 5.69753480996872523322426781716, 7.06561349234113979811058530109, 7.76494116857753110274493080363, 8.981873131895439639487253273588, 10.95054298012902263930878320015, 12.19540344222557387263325395837, 12.46954979555055655138545642500, 13.83112550727806238278890888257
