| L(s) = 1 | − 2.82i·2-s + 7.41i·3-s − 8.00·4-s + (−23.4 − 8.67i)5-s + 20.9·6-s + 42.4·7-s + 22.6i·8-s + 25.9·9-s + (−24.5 + 66.3i)10-s + 123. i·11-s − 59.3i·12-s − 165. i·13-s − 120. i·14-s + (64.3 − 173. i)15-s + 64.0·16-s − 391.·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.824i·3-s − 0.500·4-s + (−0.937 − 0.347i)5-s + 0.582·6-s + 0.866·7-s + 0.353i·8-s + 0.320·9-s + (−0.245 + 0.663i)10-s + 1.02i·11-s − 0.412i·12-s − 0.981i·13-s − 0.612i·14-s + (0.286 − 0.772i)15-s + 0.250·16-s − 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4043654785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4043654785\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 5 | \( 1 + (23.4 + 8.67i)T \) |
| 23 | \( 1 + (207. - 486. i)T \) |
| good | 3 | \( 1 - 7.41iT - 81T^{2} \) |
| 7 | \( 1 - 42.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 123. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 165. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 391.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 286. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 203.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 289.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.98e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.30e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.72e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.84e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.75e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.67e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 2.56e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.83e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.49e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.29e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.26e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 6.38e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.51e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.34e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.71223413487962203101719983457, −10.98298120997407888080478048155, −10.13755800391741829709258080926, −9.127087164208628030732331193006, −8.171955259502882021063383078906, −7.12728456617917664072653089587, −4.97861981739704113993239723043, −4.57032328344659332685573483961, −3.40373472709841029700496474724, −1.65423286850478484021549603365,
0.13704541938620845729264682202, 1.81008198089496384352776280263, 3.78069159145977806954461032204, 4.85878756248430292047341118302, 6.48952167117284026557295748084, 7.00859154229303674267587045467, 8.229996941455236942849204751548, 8.565701507356689051339929081675, 10.30207380123262580843234539500, 11.42734804361090463905375220130
