| L(s) = 1 | − 1.73·3-s − 5i·5-s − 10.3·7-s + 2.99·9-s + 10.3i·11-s − 18i·13-s + 8.66i·15-s − 10i·17-s − 13.8i·19-s + 18·21-s + 6.92·23-s − 25·25-s − 5.19·27-s − 36·29-s + 6.92i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − i·5-s − 1.48·7-s + 0.333·9-s + 0.944i·11-s − 1.38i·13-s + 0.577i·15-s − 0.588i·17-s − 0.729i·19-s + 0.857·21-s + 0.301·23-s − 25-s − 0.192·27-s − 1.24·29-s + 0.223i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3372790340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3372790340\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + 5iT \) |
| good | 7 | \( 1 + 10.3T + 49T^{2} \) |
| 11 | \( 1 - 10.3iT - 121T^{2} \) |
| 13 | \( 1 + 18iT - 169T^{2} \) |
| 17 | \( 1 + 10iT - 289T^{2} \) |
| 19 | \( 1 + 13.8iT - 361T^{2} \) |
| 23 | \( 1 - 6.92T + 529T^{2} \) |
| 29 | \( 1 + 36T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 54iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 26iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 - 41.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 90.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 90.0T + 6.88e3T^{2} \) |
| 89 | \( 1 - 18T + 7.92e3T^{2} \) |
| 97 | \( 1 - 72iT - 9.40e3T^{2} \) |
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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
−9.766107451733233691237810484595, −9.547099585895066536501879117065, −8.432184121924987137209672731281, −7.38262866526949849934702254611, −6.63036196107310203467228732836, −5.59014390618975757554875717084, −4.96666397511866811299819884950, −3.82634061070980053772342529961, −2.65792896148239648001035362452, −0.952697880073195944485700767324,
0.14230542934188463885537818562, 2.04674659355823915829253314929, 3.39854392845080213094766336230, 3.95112752236971144011578563588, 5.60467604366984907486545822971, 6.30055371753355314058630773945, 6.79254761920747454071435528452, 7.74159443176548448475384242428, 9.058305164708492763947082828343, 9.663853632848914128920388372197
