| L(s) = 1 | + (−0.622 + 8.97i)3-s − 0.562i·5-s − 55.8·7-s + (−80.2 − 11.1i)9-s − 87.7i·11-s + 71.5·13-s + (5.04 + 0.349i)15-s + 334. i·17-s − 365.·19-s + (34.7 − 501. i)21-s − 789. i·23-s + 624.·25-s + (150. − 713. i)27-s − 236. i·29-s + 1.28e3·31-s + ⋯ |
| L(s) = 1 | + (−0.0691 + 0.997i)3-s − 0.0224i·5-s − 1.13·7-s + (−0.990 − 0.138i)9-s − 0.725i·11-s + 0.423·13-s + (0.0224 + 0.00155i)15-s + 1.15i·17-s − 1.01·19-s + (0.0788 − 1.13i)21-s − 1.49i·23-s + 0.999·25-s + (0.206 − 0.978i)27-s − 0.281i·29-s + 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0691i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 + 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.264016496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264016496\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.622 - 8.97i)T \) |
| good | 5 | \( 1 + 0.562iT - 625T^{2} \) |
| 7 | \( 1 + 55.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 87.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 71.5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 334. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 365.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 789. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 236. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.28e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 513.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.18e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 204.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.31e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 317. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.27e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.77e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.24e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.98e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 838. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.39e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.57e3T + 8.85e7T^{2} \) |
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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
−10.49613420088510821425624151345, −10.01451270032996928729471404277, −8.780414731022238897201441949477, −8.368816364188796090398560122092, −6.48224295936161029525643060567, −6.07500219460967775974555949793, −4.63468108989534722560894951307, −3.67620627836158510411763413532, −2.68251060843887240361338695076, −0.49269834415615655011761355081,
0.849093468015425998021770226142, 2.32458048487894615598399578576, 3.40097007186188197708212910749, 4.98358817736894727321894752455, 6.22905965079792170922586313525, 6.86167309354928963406438237917, 7.74439693055926505975260361369, 8.906620088431279651363024007244, 9.700786642038801851071296634616, 10.81549019247788263731816408651
