| L(s) = 1 | + (−15.5 − i)3-s + (240. + 31.1i)9-s + 474i·11-s − 1.41e3i·17-s + 1.26e3·19-s − 3.12e3·25-s + (−3.71e3 − 724. i)27-s + (474 − 7.37e3i)33-s − 1.64e4i·41-s + 8.91e3·43-s + 1.68e4·49-s + (−1.41e3 + 2.20e4i)51-s + (−1.96e4 − 1.26e3i)57-s + 4.84e4i·59-s − 2.97e4·67-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0641i)3-s + (0.991 + 0.128i)9-s + 1.18i·11-s − 1.19i·17-s + 0.803·19-s − 25-s + (−0.981 − 0.191i)27-s + (0.0757 − 1.17i)33-s − 1.52i·41-s + 0.735·43-s + 49-s + (−0.0764 + 1.18i)51-s + (−0.801 − 0.0515i)57-s + 1.81i·59-s − 0.810·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0641 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0641 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9959012901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9959012901\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.5 + i)T \) |
| good | 5 | \( 1 + 3.12e3T^{2} \) |
| 7 | \( 1 - 1.68e4T^{2} \) |
| 11 | \( 1 - 474iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.64e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 8.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.84e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 + 2.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.07e9T^{2} \) |
| 83 | \( 1 - 8.92e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.49e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 8.54e4T + 8.58e9T^{2} \) |
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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
−10.74186719638626324803625487215, −9.919006966656868946918910772674, −9.164611966651275548587294962961, −7.53382161164906147588888502885, −7.10228691476581801793932003813, −5.84088195826473587930625622708, −4.99926058637496078519557836094, −4.00643561877099630690593931984, −2.30725099578983947053391483440, −0.956754037727464349758726201996,
0.35718485082229875548867360965, 1.55529597136221000885224437089, 3.32898853573931163213926122579, 4.43359940849658964561248220190, 5.69260834548156464518648493017, 6.18039074858922837520975255443, 7.41936783811344096607516252619, 8.400160401555777302243441414644, 9.568220370796130472585592724622, 10.45206608194364388914571999408
