| L(s) = 1 | − 12.3i·3-s + 21.9i·5-s + 63.3i·7-s − 71.0·9-s + 1.17·11-s − 173.·13-s + 271.·15-s + 469.·17-s + 27.8i·19-s + 780.·21-s − 79.3·23-s + 141.·25-s − 122. i·27-s − 696. i·29-s − 1.19e3·31-s + ⋯ |
| L(s) = 1 | − 1.37i·3-s + 0.879i·5-s + 1.29i·7-s − 0.877·9-s + 0.00967·11-s − 1.02·13-s + 1.20·15-s + 1.62·17-s + 0.0770i·19-s + 1.77·21-s − 0.150·23-s + 0.226·25-s − 0.167i·27-s − 0.828i·29-s − 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4189645057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4189645057\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (-1.43e3 - 1.16e3i)T \) |
| good | 3 | \( 1 + 12.3iT - 81T^{2} \) |
| 5 | \( 1 - 21.9iT - 625T^{2} \) |
| 7 | \( 1 - 63.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 1.17T + 1.46e4T^{2} \) |
| 13 | \( 1 + 173.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 469.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 27.8iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 79.3T + 2.79e5T^{2} \) |
| 29 | \( 1 + 696. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.19e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.53e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.45e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 1.69e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.99e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.56e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.44e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.93e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 9.66e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.60e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 1.14e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.52e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 2.55e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.65e3T + 8.85e7T^{2} \) |
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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
−10.15455319009835028037525951658, −9.378269107884767012917990816275, −8.155624957399794995467058043185, −7.61427738675453897962233887779, −6.72341278360664173383088146984, −5.99410034237080944344998067888, −5.07959473498383544399553037554, −3.22806622911911695076934688295, −2.45705945984204162950417838505, −1.48904117870924614759535255360,
0.098783986261198633154608917457, 1.36413412820279608786042519875, 3.25521398131995958757041934337, 4.04319283564290735405011224791, 4.89678329542046002011258600380, 5.48300313106697524129534400241, 7.09127010141081629528057238030, 7.82031621674628966308147054750, 9.001404297100736570450711041026, 9.606917662521697630322005246633
